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BYRNE'S EUCLID 



THE FIRST SIX BOOKS OF 

liTHE ELEMENTS OF EUCLID 

WITH COLOURED DIAGRAMS 
AND SYMBOLS 



.V' » 



THE FIRST SIX BOOKS OF 

THE ELEMENTS OF EUCLID 

IN WHICH COLOURED DIAGRAMS AND SYMBOLS 

ARE USED INSTEAD OF LETTERS FOR THE 

GREATER EASE OF LEARNERS 

BY OLIVER BYRNE 

SURVEYOR OF HER MAJESTY'S SETTLEMENTS IN THE FALKLAND ISLANDS 
AND AUTHOR OF NUMEROUS MATHEMATICAL WORKS 




LONDON 

WILLIAM PICKERING 

1847 



TO THE 

RIGHT HONOURABLE THE EARL FITZWILLL\M, 

ETC. ETC. ETC. 

THIS WORK IS DEDICATED 
BY HIS LORDSHIPS OBEDIENT 

AND MUCH OBLIGED SERVANT, 

OLIVER BYRNE. 



INTRODUCTION. 




HE arts and fciences have become fo extenfive, 
that to faciUtate their acquirement is of as 
much importance as to extend their boundaries. 
Illuftration, if it does not fhorten the time of 
ftudy, will at leaft make it more agreeable. This Work 
has a greater aim than mere illuftration ; we do not intro- 
duce colours for the purpofe of entertainment, or to amufe 
by certain combinations of tint and form, but to airift the 
mind in its refearches after truth, to increafe the facilities 
of inflrudlion, and to diffufe permanent knowledge. If we 
wanted authorities to prove the importance and ufefulnefs 
of geometry, we might quote every philofopher fmce the 
days of Plato. Among the Greeks, in ancient, as in the 
fchool of Peftalozzi and others in recent times, geometry 
was adopted as the befl: gymnaftic of the mind. In facfl, 
Euclid's Elements have become, by common confent, the 
bafis of mathematical fcience all over the civilized globe. 
But this will not appear extraordinary, if we confider that 
this fublime fcience is not only better calculated than any 
other to call forth the fpirit of inquiry, to elevate the mind, 
and to ftrengthen the reafoning faculties, but alfo it forms 
the beft introdudlion to moft of the ufeful and important 
vocations of human life. Arithmetic, land-furveying, men- 
furation, engineering, navigation, mechanics, hydroftatics, 
pneumatics, optics, phyfical aftronomy, &c. are all depen- 
dent on the propolitions of geometry. 



viii INTRODUCTION. 

Much however depends on the firft communication of 
any fcience to a learner, though the beft and moft eafy 
methods are feldom adopted. Propofitions are placed be- 
fore a ftudent, who though having a fufficient underftand- 
ing, is told juft as much about them on entering at the 
very threfliold of the fcience, as gives him a prepolleffion 
moft unfavourable to his future ftudy of this delightful 
fubjedl ; or " the formalities and paraphernalia of rigour are 
fo oftentatioufly put forward, as almoft to hide the reality. 
Endlefs and perplexing repetitions, which do not confer 
greater exactitude on the reafoning, render the demonftra- 
tions involved and obfcure, and conceal from the view of 
the ftudent the confecution of evidence." Thus an aver- 
fion is created in the mind of the pupil, and a fubjeft fo 
calculated to improve the reafoning powers, and give the 
habit of clofe thinking, is degraded by a dry and rigid 
courfe of inftrudlion into an uninterefting exercife of the 
memory. To raife the curiofity, and to awaken the liftlefs 
and dormant powers of younger minds fliould be the aim 
of every teacher ; but where examples of excellence are 
wanting, the attempts to attain it are but few, while emi- 
nence excites attention and produces imitation. The objedl 
of this Work is to introduce a method of teaching geome- 
try, which has been much approved of by many fcientific 
men in this country, as well as in France and America. 
The plan here adopted forcibly appeals to the eye, the moft 
fenlitive and the moft comprehenfive of our external organs, 
and its pre-eminence to imprint it fubjedl on the mind is 
fupported by the incontrovertible maxim exprefled in the 
well known words of Horace : — 

Segnius irritant animos demijfa per auran 
^uam qua fimt oculis fuhjeSla fidelibus. 
A feebler imprefs through the ear is made, 
Than what is by the faithful eye conveyed. 



INTRODUCTION. ix 

All language confifts of reprefentative figns, and thole 
figns are the befl which efFedl their purpofes with the 
greateft precifion and difpatch. Such for all common pur- 
pofes are the audible figns called words, which are ftill 
confidered as audible, whether addreffed immediately to the 
ear, or through the medium of letters to the eye. Geo- 
metrical diagrams are not figns, but the materials of geo- 
metrical fcience, the objedt of which is to Ihow the relative 
quantities of their parts by a procefs of reafoning called 
Demonftration. This reafoning has been generally carried 
on by words, letters, and black or uncoloured diagrams ; 
but as the ufe of coloured fymbols, figns, and diagrams in 
the linear arts and fciences, renders the procefs of reafon- 
ing more precife, and the attainment more expeditious, they 
have been in this inflance accordingly adopted. 

Such is the expedition of this enticing mode of commu- 
nicating knowledge, that the Elements of Euclid can be 
acquired in lefs than one third the time ufually employed, 
and the retention by the memory is much more permanent; 
thefe facts have been afcertained by numerous experiments 
made by the inventor, and feveral others who have adopted 
his plans. The particulars of which are few and obvious ; 
the letters annexed to points, lines, or other parts of a dia- 
gram are in fadt but arbitrary names, and reprefent them in 
the demonftration ; inftead of thefe, the parts being differ- 
ently coloured, are made g 
to name themfelves, for 
their forms incorrefpond- 
ing colours represent them 
in the demonftration. 

In order to give a bet- 
ter idea of this fyftem, and A 
of the advantages gained by its adoption, let us take a right 




X INTRODUCTION. 

angled triangle, and exprefs fome of its properties both by 
colours and the method generally employed. 

Some of the properties of the right angled triangle ABC, 
expreffed by the method generally employed. 

1 . The angle BAC, together with the angles BCA and 
ABC are equal to two right angles, or twice the angle ABC. 

2. The angle CAB added to the angle ACB will be equal 
to the angle ABC. 

3. The angle ABC is greater than either of the angles 
BAC or BCA. 

4. The angle BCA or the angle CAB is lefs than the 
angle ABC. 

5. If from the angle ABC, there be taken the angle 
BAC, the remainder will be equal to the angle ACB. 

6. The fquare of AC is equal to the fum of the fquares 
of AB and BC. 

The fame properties expreffed by colouring the different parts. 

That is, the red angle added to the yellow angle added to 
the blue angle, equal twice the yellow angle, equal two 
right angles. 

-^ + A = 

Or in words, the red angle added to the blue angle, equal 
the yellow angle. 





▲ 



<^H^ CZ JK^ or 

The yellow angle is greater than either the red or blue 
angle. 



INTRODUCTION. xl 



iL 




4. jl^^ or 

Either the red or blue angle is lefs than the yellow angle. 

^^^^^ minus ^HL 
In other terms, the yellow angle made lefs by the blue angle 
equal the red angle. 






That is, the fquare of the yellow line is equal to the fum 
of the fquares of the blue and red lines. 

In oral demonftrations we gain with colours this impor- 
tant advantage, the eye and the ear can be addreffed at the 
fame moment, fo that for teaching geometry, and other 
linear arts and fciences, in clafTes, the fyftem is the beft ever 
propofed, this is apparent from the examples juft given. 

Whence it is evident that a reference from the text to 
the diagram is more rapid and fure, by giving the forms 
and colours of the parts, or by naming the parts and their 
colours, than naming the parts and letters on the diagram. 
Befides the fuperior limplicity, this fyftem is likewife con- 
fpicuous for concentration, and wholly excludes the injuri- 
ous though prevalent pradlice of allowing the ftudent to 
commit the demonftration to memory ; until reafon, and fadl, 
and proof only make impreffions on the underftanding. 

Again, when ledluring on the principles or properties of 
figures, if we mention the colour of the part or parts re- 
ferred to, as in faying, the red angle, the blue line, or lines, 
&c. the part or parts thus named will be immediately feen 
by all in the clafs at the fame inftant ; not fo if we fay the 
angle ABC, the triangle PFQ^the figure EGKt, and fo on ; 



xii INTRODUCTION. 

for the letters mufl be traced one by one before the fludents 
arrange in their minds the particular magnitude referred to, 
which often occafions confufion and error, as well as lofs of 
time. Alfo if the parts which are given as equal, have the 
fame colours in any diagram, the mind will not wander 
from the objedl before it ; that is, fuch an arrangement pre- 
fents an ocular demonftration of the parts to be proved 
equal, and the learner retains the data throughout the whole 
of the reafoning. But whatever may be the advantages of 
the prefent plan, if it be not fubftituted for, it can always 
be made a powerful auxiliary to the other methods, for the 
purpofe of introdudlion, or of a more fpeedy reminifcence, 
or of more permanent retention by the memory. 

The experience of all who have formed fyftems to im- 
prefs fadts on the underftanding, agree in proving that 
coloured reprefentations, as pidlures, cuts, diagrams, &c. are 
more eafily hxed in the mind than mere fentences un- 
marked by any peculiarity. Curious as it may appear, 
poets feem to be aware of this fadl more than mathema- 
ticians ; many modern poets allude to this viiible fyftem of 
communicating knowledge, one of them has thus expreffed 
himfelf : 

Sounds which addrefs the ear are loft and die 
In one fhort hour, but thefe which ftrilce the eye, 
Live long upon the mind, the faithful fight 
Engraves the knowledge with a beam of light. 

This perhaps may be reckoned the only improvement 
which plain geometry has received fince the days of Euclid, 
and if there were any geometers of note before that time, 
Euclid's fuccefs has quite eclipfed their memory, and even 
occalioned all good things of that kind to be alfigned to 
him ; like ^Efop among the writers of Fables. It may 
alfo be worthy of remark, as tangible diagrams afford the 
only medium through which geometry and other linear 



INTRODUCTION. xiii 

arts and fciences can be taught to the blind, this vifible fys- 
tem is no lefs adapted to the exigencies of the deaf and 
dumb. 

Care muft be taken to fliow that colour has nothing to 
do with the lines, angles, or magnitudes, except merely to 
name them. A mathematical line, which is length with- 
out breadth, cannot poffefs colour, yet the jundtion of two 
colours on the fame plane gives a good idea of what is 
meant by a mathematical line ; recolledt we are fpeaking 
familiarly, fuch a jundlion is to be underftood and not the 
colour, when we fay the black line, the red line or lines, &c. 

Colours and coloured diagrams may at firfl: appear a 
clumiy method to convey proper notions of the properties 
and parts of mathematical figures and magnitudes, how- 
ever they will be found to afford a means more refined and 
extenfive than any that has been hitherto propofed. 

We fliall here define a point, a line, and a furface, and 
demonflrate a propofition in order to fhow the truth of this 
affertion. 

A point is that which has pofition, but not magnitude ; 
or a point is pofition only, abftradled from the confideration 
of length, breadth, and thicknefs. Perhaps the follow- 
ing defcription is better calculated to explain the nature of 
a mathematical point to thofe who have not acquired the 
idea, than the above fpecious definition. 

Let three colours meet and cover a 
portion of the paper, where they meet 
is not blue, nor is it yellow, nor is it 
red, as it occupies no portion of the 
plane, for if it did, it would belong 
to the blue, the red, or the yellow 
part; yet it exifts, and has pofition 
without magnitude, fo that with a Uttle refledlion, this June- 




XIV 



INTRODUCTION. 




tion of three colours on a plane, gives a good idea of a 
mathematical point. 

A line is length without breadth. With the afliftance 
of colours, nearly in the fame manner as before, an idea of 
a line may be thus given : — 

Let two colours meet and cover a portion of the paper; 

where they meet is not red, nor is it 
blue ; therefore the jundlion occu- 
pies no portion of the plane, and 
therefore it cannot have breadth, but 
only length : from which we can 
readily form an idea of what is meant by a mathematical 
line. For the purpofe of illuftration, one colour differing 
from the colour of the paper, or plane upon which it is 
drawn, would have been fufficient ; hence in future, if we 
fay the red line, the blue line, or lines, &c. it is the junc- 
tions with the plane upon which they are drawn are to be 
underftood. 

Surface is that which has length and breadth without 
thicknefs. 

When we confider a folid body 
(PQ), we perceive at once that it 
has three dimenfions, namely : — 
length, breadth, and thicknefs ; 
fuppofe one part of this folid (PS) 
to be red, and the other part (QR) 
yellow, and that the colours be 
diflinft without commingling, the 
blue furface (RS) which feparates 
thefe parts, or which is the fame 
S thing, that which divides the folid 
without lofs of material, mufl be 
without thicknefs, and only poffeffcs length and breadth ; 






INTRODUCTION. 



XV 



this plainly appears from reafoning, limilar to that juft em- 
ployed in defining, or rather delcribing a point and a line. 

The propofition which we have felefted to elucidate the 
manner in which the principles are applied, is the fifth of 
the firft Book. 

In an ifofceles triangle ABC, the 
internal angles at the bafe ABC, 
ACB are equal, and when the fides 
AB, AC are produced, the exter- 
nal angles at the bafe BCE, CBD 
are allb equal. 

Produce _i__ and 
make ■■■■ "^ 

Draw ^— — and 
(B. i.pr. 3.) 



and 





and 






common 



and 



^ = -^ (B. I. pr. 4.) 
Again in >^ and N. t ^ 



xvi INTRODUCTION. 



and ^ = ^; 

and ^^^ ^ ^^^ (B. i. pr. 4). 



But 

C^E. D. 

By annexing Letters to the Diagratn. 

Let the equal fides AB and AC be produced through the 
extremities BC, of the third Tide, and in the produced part 
BD of either, let any point D be afllimed, and from the 
other let AE be cut off equal to AD (B. i. pr. 3). Let 
the points E and D, fo taken in the produced fides, be con- 
nedted by ftraight lines DC and BE with the alternate ex- 
tremities of the third fide of the triangle. 

In the triangles DAC and EAB the fides DA and AC 
are refpedlively equal to EA and AB, and the included 
angle A is common to both triangles. Hence (B i . pr. 4.) 
the line DC is equal to BE, the angle ADC to the angle 
AEB, and the angle ACD to the angle ABE ; if from 
the equal lines AD and AE the equal fides AB and AC 
be taken, the remainders BD and CE will be equal. Hence 
in the triangles BDC and CEB, the fides BD and DC are 
refpedively equal to CE and EB, and the angles D and E 
included by thofe fides are alfo equal. Hence (B. i. pr. 4.) 



INTRODUCriON. xvii 

the angles DBC and ECB, which are thofe included by 

the third fide BC and the productions of the equal fides 

AB and AC are equal. Alfo the angles DCB and EBC 

are equal if thofe equals be taken from the angles DCA 

and EBA before proved equal, the remainders, which are 

the angles ABC and ACB oppofite to the equal fides, will 

be equal. 

Therefore in aii ifofceles triangle y &c. 

Q^E. D. 

Our object in this place being to introduce the fyftem 
rather than to teach any particular fet of propofitions, we 
have therefore feledled the foregoing out of the regular 
courfe. For fchools and other public places of infi:rud:ion, 
dyed chalks will anfwer to defcribe diagrams, &c. for private 
ufe coloured pencils will be found very convenient. 

We are happy to find that the Elements of Mathematics 
now forms a confiderable part of every found female edu- 
cation, therefore we call the attention of thofe interefiied 
or engaged in the education of ladies to this very attractive 
mode of communicating knowledge, and to the fucceeding 
work for its future developement. 

We fhall for the prefent conclude by obferving, as the 
fenfes of fight and hearing can be fo forcibly and infiianta- 
neously addreffed alike with one thoufand as with one, the 
million might be taught geometry and other branches of 
mathematics with great eafe, this would advance the pur- 
pofe of education more than any thing that might be named, 
for it would teach the people how to think, and not what 
to think ; it is in this particular the great error of education 
originates. 



XVlll 



THE ELEMENTS OF EUCLID. 
BOOK I. 

DEFINITIONS. 

I. 

A point is that which has no parts. 

II. 

A line is length without breadth. 

III. 

The extremities of a line are points. 

IV. 

A ftraight or right line is that which lies evenly between 

its extremities. 

V. 

A furface is that which has length and breadth only. 

VI. 

The extremities of a furface are lines. 

VII. 

A plane furface is that which lies evenly between its ex- 
tremities. 

VIII. 

A plane angle is the inclination of two lines to one ano- 
ther, in a plane, which meet together, but are not in the 

fame diredlion. 

IX. 

^ A plane redlilinear angle is the inclina- 

^r tion of two ftraight lines to one another, 

^^^ which meet together, but are not in the 

ir fame flraight line. 



BOOK I. DEFINITIONS. 



XIX 



When one ftraight line Handing on ano- 
ther ftraight Hne makes the adjacent angles 
equal, each of thefe angles is called a rigkf 
angle, and each of thefe lines is faid to be 
perpendicular to the other. 



A 





XI. 

An obtufe angle is an angle greater 
than a right angle. 

XII. 

An acute angle is an angle lefs than a 
right angle. 

XIII. 
A term or boundary is the extremity of any thing. 

XIV. 

A figure is a furface enclofed on all fides by a line or lines. 

XV. 

A circle is a plane figure, bounded 
by one continued line, called its cir- 
cumference or periphery ; and hav- 
ing a certain point within it, from 
which all ftraight lines drawn to its 
circumference are equal. 

XVI. 

This point (from which the equal lines are drawn) is 
called the centre of the circle. 




XX BOOK I. DEFINITIONS. 



XVII. 
A diameter of a circle is a ftraight line drawn 
through the centre, terminated both ways 
in the circumference. 

XVIII. 

A femicircle is the figure contained by the 
diameter, and the part of the circle cut off 
by the diameter. 

XIX. 

A fegment of a circle is a figure contained 
by a ftraight line, and the part of the cir- 
cumference which it cuts off. 




^•••••••* 




••'•' 



XX. 

A figure contained by ftraight lines only, is called a redli- 
linear figure. 

XXI. 
A triangle is a redlilinear figure included by three fides. 

XXII. 

A quadrilateral figure is one which is bounded 
by four fides. The fi:raight lines ■^— «— . 
and .^_«— i«> connecfting the vertices of the 
oppofite angles of a quadrilateral figure, are 
called its diagonals. 

XXIII. 

A polygon is a redilinear figure bounded by more than 
four fides. 




BOOK I. DEFINITIONS. 



XXI 





XXIV. 

A triangle whofe three fides are equal, is 
faid to be equilateral. 

XXV. 

A triangle which has only two fides equal 
is called an ilbfceles triangle. 

XXVI. " 

A fcalene triangle is one which has no two fides equal. 

XXVII. 

A right angled triangle is that which 
has a right angle. 

XXVIII. 

An obtufe angled triangle is that which 
has an obtufe angle. 

XXIX. 

An acute angled triangle is that which 
has three acute angles. 

XXX. 

Of four-fided figures, a fquare is that which 

has all its fides equal, and all its angles right 

angles. 

XXXI. 

A rhombus is that which has all its fides 
equal, but its angles are not right angles. 

XXXII. 




u 



An oblong is that which has all its 
angles right angles, but has not all its 
fides equal. 



xxii BOOK L POS'lVLATES. 



XXXIII. 

A rhomboid is that which has its op- 
pofite fides equal to one another, 
but all its fides are not equal, nor its 



angles right angles. 

XXXIV. 

All other quadrilateral figures are called trapeziums. 

XXXV, 

^^—--^,^g„^^^ Parallel ftraight lines are fuch as are in 
^^^^^^^^^^ the fame plane, and which being pro- 
duced continually in both directions, 
would never meet. 

POSTULATES. 
I. 

Let it be granted that a flraight line may be drawn from 
any one point to any other point. 

II. 

Let it be granted that a finite ftraight line may be pro- 
duced to any length in a ftraight line. 

III. 
Let it be granted that a circle may be defcribed with any 
centre at any diflance from that centre. 

AXIOMS. 
I. 

Magnitudes which are equal to the fame are equal to 

each other. 

II. 

If equals be added to equals the fums will be equal. 



BOOK I. AXIOMS. xxiii 

III. 

If equals be taken away from equals the remainders will 

be equal. 

IV. 

If equals be added to unequals the fums will be un- 
equal. 

V. 

If equals be taken away from unequals the remainders 

will be unequal. 

VI. 

The doubles of the fame or equal magnitudes are equal. 

VII. 

The halves of the fame or equal magnitudes are equal. 

VIII. 

Magnitudes which coincide with one another, or exactly 
fill the fame fpace, are equal. 

IX. 

The whole is greater than its part, 

X. 

Two ftraight lines cannot include a fpace. 

XI. 

All right angles are equal. 

XII. 

If two ftraight lines ( } meet a third 

ftraight line ( ) fo as to make the two interior 

angles ( and jj^ ) on the fame fide lefs than 

two right angles, thefe two ftraight lines will meet if 
they be produced on that fide on which the angles 
are lefs than two right angles. 




XXIV 



BOOK I. ELUCIDATIONS. 



The twelfth axiom may be expreffed in any of the fol- 
lowing ways : 

1 . Two diverging ftraight lines cannot be both parallel 
to the fame flraight line. 

2. If a ftraight line interfeft one of the two parallel 
ftraight lines it mufl alfo interfedt the other. 

3. Only one ftraight line can be drawn through a given 
point, parallel to a given ftraight line. 

Geometry has for its principal objefts the expofition and 

explanation of the properties oi figure, and figure is defined 

to be the relation which fubfifts between the boundaries of 

fpace. Space or magnitude is of three kinds, linear, fuper- 

ficial, ■Si.w^foUd. 

Angles might properly be confidered as a fourth fpecies 
of magnitude. Angular magnitude evidently confifts of 
parts, and muft therefore be admitted to be a fpecies ol 
quantity The ftudent muft not fuppofe that the magni- 
tude of an angle is affefted by the length 
of the ftraight lines which include it, and 
of whofe mutual divergence it is the mea- 
fure. The vertex of an angle is the point 
where \}[\& fides or the legs of the angle 
meet, as A. 
An angle is often defignated by a fingle letter when its 
legs are the only lines which meet to- 
gether at its vertex. Thus the red and 
blue lines form the yellow angle, which 
in other fyftems would be called the 
angle A. But when more than two 
B lines meet in the fame point, it was ne- 
ceflary by former methods, in order to 
avoid confufion, to employ three letters 
to defignate an angle about that point. 





BOOK I. ELUCIDATIONS. xxv 

the letter which marked the vertex of the angle being 
always placed in the middle. Thus the black and red lines 
meeting together at C, form the blue angle, and has been 
ufually denominated the angle FCD or DCF The lines 
FC and CD are the legs of the angle; the point C is its 
vertex. In like manner the black angle would be defignated 
the angle DCB or BCD. The red and blue angles added 
together, or the angle HCF added to FCD, make the angle 
HCD ; and fo of other angles. 

When the legs of an angle are produced or prolonged 
beyond its vertex, the angles made by them on both fides 
of the vertex are faid to be vertically oppofite to each other : 
Thus the red and yellow angles are faid to be vertically 
oppofite angles. 

Superpojition is the procefs by which one magnitude may 
be conceived to be placed upon another, fo as exadlly to 
cover it, or fo that every part of each fhall exadly coin- 
cide. 

A line is faid to be produced, when it is extended, pro- 
longed, or has its length increafed, and the increafe of 
length which it receives is called its produced part, or its 
produSlion. 

The entire length of the line or lines which enclofe a 
figure, is called its perimeter. The firft fix books of Euclid 
treat of plain figures only. A line drawn from the centre 
of a circle to its circumference, is called a radius. The 
lines which include a figure are called \isjides. That fide 
of a right angled triangle, which is oppofite to the right 
angle, is called the hypotenufe. An oblong is defined in the 
fecond book, and called a reSlangle. All the lines which 
are confidered in the firfl: fix books of the Elements are 
fuppofed to be in the fame plane. 

The Jiraight-edge and compajfcs are the only inflruments. 



xxvi BOOK I. ELUCIDATIONS. 

the ufe of which is permitted in Euclid, or plain Geometry. 
To declare this reflridlion is the objedl of the pojiulates. 

The Axioms of geometry are certain general proportions, 
the truth of which is taken to be felf-evident and incapable 
of being eftabliflied by demonftration. 

Propojitions are thofe refults which are obtained in geo- 
metry by a procefs of reafoning. There are two fpecies of 
propofitions in geometry, problems and theorems. 

A Problem is a propofition in which fomething is pro- 
pofed to be done ; as a line to be drawn under fome given 
conditions, a circle to be defcribed, fome figure to be con- 
rtrudled, &c. 

Th.t folution of the problem confifts in fhowing how the 
thing required may be done by the aid of the rule or ftraight- 
edge and compafTes. 

The demonftration confifts in proving that the procefs in- 
dicated in the folution really attains the required end. 

A Theorem is a propofition in which the truth of fome 
principle is afi^erted. This principle mufl: be deduced from 
the axioms and definitions, or other truths previously and 
independently ellabliihed. To fhow this is the objedl of 
demonftration. 

A Problem is analogous to a poftulate. 

A Theorem refembles an axiom. 

A Pojlulate is a problem, the folution of which is afiiimed. 

An Axiom is a theorem, the truth of which is granted 
without demonftration. 

A Corollary is an inference deduced immediately from a 
propofition. 

A Scholium is a note or obfervation on a propofition not 
containing an inference of fufiicient importance to entitle it 
to the name of a corollary. 

A Lemma is a propofition merely introduced for the pur- 
pofe of eftabliftiing fome more important propofition. 



xxvu 

SYMBOLS AND ABBREVIATIONS. 

,*, exprefles the word therefore. 

*,' becaufe. 

zz equal. This fign of equaHty may 

be read equal to, or is equal to, or are equal to ; but 
any difcrepancy in regard to the introdudlion of the 
auxiliary verbs Is, are, &c. cannot affedl the geometri- 
cal rigour. 

^ means the fame as if the words ' not equal' were written. 

r~ fignifies greater than. 

^ . . . . lefs than. 

Cjl . . . . not greater than. 

j] . . . . not lefs than. 

-\- is vtzdplus [fjiore), the fign of addition ; when interpofed 
between two or more magnitudes, fignifies their fum. 

— is read minus {lefs), fignifies fubtracftion ; and when 
placed between two quantities denotes that the latter 
is to be taken from the former. 

X this fign exprefi"es the produdl of two or more numbers 
when placed between them in arithmetic and algebra ; 
but in geometry it is generally ufed to exprefs a rect- 
angle, when placed between " two flraight lines which 
contain one of its right angles." A reBangle may alfo 
be reprefented by placing a point between two of its 
conterminous fides. 

: :; : exprefies an analogy or proportion ; thus, if A, B, C 
and D, reprefent four magnitudes, and A has to 
B the fame ratio that C has to D, the propofition 
is thus briefly written, 

A : B ; : C : D, 
A : B = C : D, 
A C 

°'"b = d. 

This equality or famenefs of ratio is read, 



xxviii STMBOLS AND ABBREVIAnONS. 

as A is to B, fo is C to D ; 

or A is to B, as C is to D. 
II fignifies parallel to. 
J_ . . . . perpendicular to. 

. angle. 

. right angle. 




CIS 



two right angles, 

^1^ or I N briefly defignates a point. 

C =, or ^ fignifies greater, equal, or lefs than. 

The fquare defcribed on a line is concifely written thus. 

In the fame manner twice the fquare of, is expreffed by 
2 \ 

def. fignifies definition. 

pos pofiulate. 

ax axiom. 

hyp hypothefis. It may be necefiary here to re- 
mark, that the hypothefis is the condition aflumed or 
taken for granted. Thus, the hypothefis of the pro- 
pofition given in the Introduction, is that the triangle 
is ifofceles, or that its legs are equal. 

conft confiruElion. The confiruBion is the change 

made in the original figure, by drawing lines, making 
angles, defcribing circles, &c. in order to adapt it to 
the argument of the demonfi:ration or the folution of 
the problem. The conditions under which thefe 
changes are made, are as indisputable as thofe con- 
tained in the hypothefis. For infi:ance, if we make 
an angle equal to a given angle, thefe two angles are 
equal by conftrudlion. 

Q^E. D ^lod erat detnonfirandum. 

Which was to be demonftrated. 



CORRIGENDA. xxix 



Faults to be correEied before reading this Volu7Jie. 

Page 13, line 9, /or def. 7 read ^z.L 10. 
45, laft line, /or pr. 19 r^^^ pr. 29. 

54, line 4 from the bottom, /or black and red line read blue 
and red line. 

59, line 4, /or add black line fquared read add blue line 
fquared. 

60, line 17, /or red line multiplied by red and yellow line 

read red line multiplied by red, blue, and yellow line. 
76, line 11, for def. 7 read dt?. 10. 
81, line lOyfor take black line r^i2ii take blue line. 
105, line 11, for yellow black angle add blue angle equal red 

angle read yellow black angle add blue angle add red 

angle. 

129, laft line, /or circle read triangle. 

141, line I, /or Draw black line read Draw blue line. 

196, line 3, before the yellow magnitude infert M. 



(Euclib. 




BOOK I. 
PROPOSITION I. PROBLEM. 



N a given finite 

firaight line ( ) 

to dejcribe an equila- 



teral triangle. 



Defcribe I "^^ and 



o 




(postulate 3.); draw and — (poft. i.). 

then will \ be equilateral. 





(def. 15.); 




— (def. 15.), 


• ^_ -mm 


— (axiom, i .) ; 


and therefore \^ is the equilateral triangle required. 




Q^E. D 



B 



BOOK I. PROP. II. PROB. 





ROM aghenp'jhit ( ■■ ), 
to draic ajiraight line equ.al 
to a green finite firaight 

line ( ). 



Draw — — — — (poil. I.), defcribe 

Afpr. I.), produce — — (poll. 

o 



2.), defcribe 



(poft. 3.), and 






(poll. 3.) ; produce — ^— "" (port. 2.), ther 
is the line required. 



For 



and 



(def. 15.), 



(conll.), .*. 



(ax. 3.), but (def. 15.' 



drawn from the given point ( 



is equal the given line 



Q. E. D. 



BOOK I. PROP. in. PROP. 



ROM the greater 

( "—) of 

tivo given Jiraight 

lines, to cut off a part equal to 

the kfs ( ). 





Draw 




(poll:. 3 .), then 



(pr. 2.) ; defcribe 



For 
and 



(def. 15.), 
(conll.) ; 
(ax. I.). 



Q. E. D. 



BOOK I. PROP. IF. THEOR. 





F two triangles 

have two fides 

of the one 

refpeSlively 

equal to two fdes of the 

other, ( I to ' 

and ^__ to w^^m. ) and 

the angles { and ^ ) 

contained by thofe equal 
fdes alfo equal ; then their bafes or their fdes (-^-^— and 
^^^^) are alfo equal : and the remaining and their remain- 
ing angles oppofte to equal fides are refpeSlively equal 

( ^^ =: ^^ and ^^ n ^^ ) ; and the triangles are 
equal in every refpeB. 

Let the two triangles be conceived, to be fo placed, that 
the vertex of the one of the equal angles. 



or 



fliall fall upon that of the other, and 



with 



then will 



^^— to coincide 
coincide with » i if ap- 



plied: confequently 



will coincide with 



or two flraight lines will enclofe a fpace, which is impoffible 



(ax. lo), therefore 



and 



^=» 



^ ^^ , and as the triangles 



* = > 



A-^ 



coincide, when applied, they are equal in every refpedl. 

Q. E. D. 




BOOK I. PROP. V. THEOR. 



N anj ifofceles triangle 



A 



if the equal Jides 
be produced, the external 
angles at the bafe are equal, and the 
internal angles at the bafe are alfo 
equal. 



Produce 



and 



y (poft. 2.), take 

— - = 9 (pr- 3-); 

draw -i^— — » and n . 




Then in 



both, and 



A A 

/ \ and / \ we have, 

= (conft.), A 



common to 



(hyp.) /. Jk = 
and ^ = ^ (pr. 4.). 




^ = ^ and 

1^^ -zz ^^ \ and ^^» ^ ^^ (pr. 4.) but 

^ = ^ "*' Jk = JL ^'-^' 



Q. E. D. 



BOOK I. PROP. Ft. THEOR. 





A 



and 



N any triangle ( / \ ) ;/' 
two angles ( ' and ^L ) 

are equal, the Jides ( ■— ■ 

■~ ) oppofite to them are alfo 



equal. 

For if the fides be not equal, let one 
of them I — ■ be greater than the 



other 



and from it cut off 



(pr. 3.), draw- 



Then 



(conft.) 



m 



A.naA, 



(hyp.) 



anc 



common, 



,*. the triangles are equal (pr. 4.) a part equal to the whole, 

which is abfurd ; ,*, neither of the fides — "» or 

' is greater than the other, /. hence they are 



equal 



Q^E. D. 



BOOK I. PROP. FII. THEOR. 




N the fame bafe (■ 



■), a7id on 



the fa}7ie Jide of it there cannot be tivo 
triangles having their conterminous 
fides ( and — ^— ^ 

•— — ■ and «i^i— ii^—) at both extremities of 
the bafe, equal to each other. 

When two triangles ftand on the fame bale, 
and on the fame iide of it, the vertex of the one 
Ihall either fall outlide of the other triangle, or 
within it ; or, laftly, on one of its lides. 



llructed fo that 



#='' 



If it be poffible let the two triangles be con- 

'«■ rzzzz — zizzz f ^^^" 

draw ——---- and, 

= ^ (Pr- 5-) 

.'. ^^ ^ ^^ and 

but (pr. 5.) yf = ^^ 

therefore the two triangles cannot have their conterminous 




which is abfurd. 



fides equal at both extremities of the bafe. 



Q. E. D. 



BOOK I. PROP. Fill. THEOR. 





F two triangles 

have two Jides 

of the one refpec- 

tjvely equal to 

two Jides of the other 

and .—m^ =r ), 

and alfo their bafes ( 

^ •— ), equal ; then the 



and 



) 



angles ( 

contained by their equal Jides 
are alfo equal. 



If the equal bafes 



and 



be conceived 



to be placed one upon the other, fo that the triangles fhall 
lie at the fame fide of them, and that the equal fides 
«______ and .i.....i_ , —«-.—. and _____ be con- 
terminous, the vertex of the one mufi: fall on the vertex 
of the other ; for to fuppofe them not coincident would 
contradidl the laft propofition. 



Therefore the fides 
cident with 



and . 
, and 



., being coin- 



k-k 



Q. E. D. 



BOOK I. PROP. IX. PROP. 




Take 



O bifeB a given reSlilinear 
angle {^ J. 



(PJ*- 3-) 



draw 



, upon which 



defcribe ^^ (pr. i.). 



draw 




Becaufe _ = ..^... (confl.) 
and ^^^— common to the two triangles 



and 



(conft.). 



4 



= (pr. 8.) 



Q. E. D. 



10 



BOOK I. PROP. X. PROB. 



O i>tye^ a given finite Jlraight 

line [f^^^mmmmwm'^. 




and 



common to the two triangles. 



Therefore the given line is bifefted. 



Q;E. D. 



BOOK L PROP. XL PROB. 



II 




( : 

a perpendicular. 



ROM a given 

point ( I ), 

in a given 

Jlraight line 

— ), to draw 




Take any point (• 
cut off 



) in the given line, 
(pr- 3-)' 



/ \ (Pr. I.), 



conftrudl 

draw — — and it fliall be perpendicular to 
the given line. 



For 



(conft.) 



(conft.) 



and 



- common to the two triangles. 




Therefore ^|| z:z. 
J. 




(pr. 8.) 
(def. 10.). 



C^E.D. 



12 



BOOK I. PROP, XII. PROB. 





O draw a 

Jlraight line 

perpendicular 

to a given 

/ indefinite Jlraight line 

(^^^ ^ from a given 

[point /ys. ) "without. 



With the given point /|\ as centre, at one fide of the 

line, and any diftance — ^^— capable of extending to 

the other fide, defcribe 



Make 
draw ^ 



(pr. 10.) 



and 



then 



For (pr. 8.) lince 



(conft.) 



and 




common to both, 
= (def. 15.) 



and 




(def. 10.). 

Q. E. D. 



BOOK I. PROP. XIII. THEOR. 



13 



HEN a Jlralght line 
( ..m^^m^ ) Jlanding 
upon another Jlraight 
line ( ) 

makes angles with it; they are 
either two right angles or together 
equal to two right angles. 





If 



be _L to 



gf..A=C£^ 



then, 

(def. 7.). 



But if 
draw 



be not _L to — — — 
J. ;(pr. II.) 



(conft.). 






Q. E. D. 



H 



BOOK I. PROP. XIV. THEOR. 




F two Jiraight lines 

fneeting a thirdjlraight 

line (i ' ), at the 

fame pointy and at oppofite Jides of 

it, make with it adjacent angles 




and 



A 



) egual to 



two right angles ; thefe fraight 
lines lie in one continuous Jiraight 
line. 



For, if pofTible let 



and not 



be the continuation of 



then 




+ 



but by the hypothefis 



4 = ^ 



+ 



(ax. 3.) ; which is abfurd (ax. 9.). 



, is not the continuation of 



and 



the like may be demonftrated of any other flraight line 
except , ,*, ^-^— is the continuation 



of 



Q. E. D. 



BOOK I. PROP. XV. THEOR. 



15 




gles 
and 



F two right lines ( 
and ■' ' I ) interfe£t one 
another, the vertical an- 



and 



^ 





are 



equal. 




► - 

<* 








► 4 



In the fame manner it may be fliown that 





Q^E. D. 



i6 




BOOK I. PROP. XVI. THEOR. 

F a fide of a 




is produced, the external 



trian- 




greater than either of the 
internal remote angles 



( 



▲ .A 



)• 



Make 



Draw 



— (pr. lo.). 

- and produce it until 
■^^— ; draw — ^— • , 



In 




and #•••' 




► 4 



and 



(conft. pr. 15.), /. ^m = ^L (pr. 4.), 

...f^.A. 

In like manner it can be fhown, that if •—-■•• 
be produced, ^^^^ Q ^^k , and therefore 

is [= ^ii. 
Q. E. D. 




which is ^z 




BOOK I. PROP. XVII. THEOR. 



17 




NY tivo angles of a tri- 
angle f * are to- 



gether lefs than two right angles. 




Produce 



+ 



then will 





^Oi 



But, mik [= Mk (pr- 16.) 

and in the fame manner it may be Ihown that any other 
two angles of the triangle taken together are lefs than two 
right angles. 

Q;E. D. 



i8 



BOOK I. PROP. XVIIL THEOR. 




A 




N any triangle 

if one Jide vbm* be 

greater than another 

•^^mmmm-^ ^ the aUgk Of- 

pojite to the greater Jide is greater 
than the angle oppoftte to the lefs. 



1. e. 



^ 



Make 



Then will 



(pr. 3.), draw 



A.A 



(pr- 5-) J 



but 





i£k 



(pr. 16.) 



and much more 



IS 



^-> 



Q. E. D. 



BOOK I. PROP. XIX. THEOR. 



19 




A 



F m any triangle 

one angle J/j^ be greater 




than another ^^^ the Jide 
which is oppojite to the greater 



angle, is greater than the Jide 
oppojite the lefs. 




If 



be not greater than 



or 



then muft 



If 



then 




which is contrary to the hypothefis. 
— is not lefs than •^■— ^—j for if it were, 

which is contrary to the hypothefis : 



Q. E. D. 



20 



BOOK I. PROP. XX. THEOR. 





NY two fides 
and iBMMH 



of a 



triangle 



Z\ 



taken together are greater than the 
third fide ( ). 



Produce 



and 



make >< 



(pr- 3-); 



draw 



Then becaufe ------ ^ 



(conft.). 






(ax. 9.) 



+ 



and ,*, 



+ 



(pr. 19.) 



Q.E.D 



BOOK I. PROP. XXL THEOR. 



21 




F from any point ( / ) 

within a triangle 

' Jlraight lines be 
drawn to the extremities of one fide 
( ), thefe lines mujl he toge- 
ther lefs than the other two fdes, but 
muJl contain a greater angle. 





Produce 



+ 



add 



to each. 



(pr. 20.), 



+ 



+ 



(ax. 4.) 



In the fame manner it may be fhown that 
... + [Z + 



which was to be proved. 




4 
■.A 



(pr. 16.), 



(pr. 16.), 




Q^E.D. 



22 



BOOK I. PROP. XXII. THEOR. 





\IVE'N three ng/it 

lines < -■••— 

the fum of any 
two greater than 
the third, to conJlru6i a tri- 
angle whofe Jides Jhall be re- 
fpeSlively equal to the given 
lines. 



■■■•■«a««^«M 



AfTume 



Draw — — ^ 

and -^— • s: 



With 



defcribe 




and 



and 







I (pr. 2.). 
as radii, 

(poft. 3.); 



draw and 



then will 



For 



and ■ 



be the triangle required. 
"' i 

Q. E. D. 



BOOK I. PROP. XXIII. PROB. 23 




iT a given point ( ) in a 

given Jiraight line (^^^»»— ■), 
to make an angle equal to a 

given re 51 i lineal angle (.^^j^ )• 

Draw — — — . between any two points 
in the legs of the given angle. 



Conftruct v (pr. 22.) 

fo that — ^^^ = . 






and 



Then jgj^ = ^J^ (pr. 8.). 

Q. E. D. 



24 



BOOK I. PROP. XXir. THEOR. 





X> 



F two triangles 
have two fides of 
the one refpec- 
tively equal to 
twofdes of the other ( 

to and ------ 

to ), and if one of 

A 

the angles ( <3. .\ ) contain- 
ed by the equal fdes be 



greater than the other (c.»«^), the fide ( ^-^-^^ ) isohich is 
oppofte to the greater angle is greater than thefde ( - . . . ) 
which is oppofte to the lefs angle. 



Make 
and — 



L^ - ly (pr. 23.), 
= (pr- 3-). 



draw ..-••-■-•» and -■——■. 
Becaufe ^— — ^ 3: — •— — (ax. i. hyp. conft.) 



but 



and .*. 



^ = ^ (F- 




but 



(pr. 19.) 
(pr.4.) 



Q. E. D. 



BOOK I. PROP. XXV. THEOR. 



25 




F two triangles 
have two fides 
( " '■ " ■' and 
) of the 



one refpeBively equal to two 

fides ( and — — ) 

of the other, but their bafes 
unequal, the angle fubtended 
by the greater bafe (««—■—■) 
of the one, muji be greater 
than the angle fubtended by 
the lefs bafe ("■"■■"*•) of the other. 





^Im- ^ , C or H] ^^ ^^^ is not equal to ^^ 
^^ •=. ^^ then ^^^^ := — — i- (pr. 4.) 



for if 



which is contrary to the hypothefis ; 
^H^ is not lefs than ^^ 

for if A :ti A 

then i "H ' (pr. 24.), 

which is alfo contrary to the hypothefis : 

/.A [= A. 



Q^E. D. 



26 BOOK I. PROP. XXVL THEOR. 

Case I. 




F two triangles 

have two angles 

of the one re- 

fpedlively equal 

to two angles of the other. 




( 



and 



Case II. 




tf) 



Let 



y), and a fide 
of the one equal to afde of 
the other fmilarly placed 
with refpeSl to the equal 
angles, the remaining fdes 
and angles are refpeSlively 
equal to one another. 



CASE I. 
and I which lie between 



the equal angles be equal, 
then -^— — ^ ^^— ■••• 



For if it be poflible, let one of them -i 
greater than the other ; 



be 



In X \ and X ^ 



we have 



M = A 



(pr.4.) 



BOOK I. PROP. XXVI. THEOR. 27 

but A = iH (hyp.) 

and therefore g^^ =: ^|B, which is abfurd ; 

hence neither of the fides — ^— ■— and — ■^■■■- is 

greater than the other; and .*. they are equal; 



and 4 = 4, 



(pr. 4.). 



CASE II. 
Again, let ^— — • ^ ■— — — ^ which lie oppofite 



the equal angles flik and ^^^ . If it be poflible, let 

Then in ' ^ and J^^^ we have 

= and /^ = J^, 



I'ut H^ = JBi^ (hyp.) 
.*. jf^ = ^^^ which is abfurd (pr. 16.). 

Confequently, neither of the fides ^"i— i"«» or ^-^"i—^ is 

greater than the other, hence they muft be equal. It 

follows (by pr. 4.) that the triangles are equal in all 

refpedls. 

Q^E. D. 



28 



BOOK I. PROP. XXVII. THEOR. 




F ajlralght line 

( ) meet- 

i?2g tivo other 

Jiraight lines, 

- and ) makes 



with them the alternate 



angles ( 




and 



) equal, thefe two Jiraight lines 



are parallel. 



If 



be not parallel to 



they fliall meet 



when produced. 



If it be poflible, let thofe lines be not parallel, but meet 
when produced ; then the external angle ^^ is greater 

than flHik>^ (pr. i6),but they are alfo equal (hyp.), which 
is abfurd : in the fame manner it may be ihown that they 
cannot meet on the other fide ; ,*, they are parallel. 

Q. E. D. 



BOOK I. PROP. XXFIIL THEOR. 



29 




(- 



F ajlraight line 

ting two other 
Jlraight lines 



makes the external equal to 
the internal and oppojite 
angle, at the fame Jide of 
the cutting line {namely. 





yl, or if it makes the two internal angles 

at the fame ftde ( ^l^ and ^F , or f/^ and ^^^) 
together equal to two right angles, thofe two Jlraight lines 
are parallel. 



Firft, if 



1^ =^^ , then Jjj^ = ^r (pr. i 
mL = W /. II (pr. 27.). 



Secondly, if 



then 





+ 








(pr. 13.), 
(ax. 3.) 



(pr. 27.) 

C^E. D. 



30 



BOOK I. PROP. XXIX. THEOR. 





STRAIGHT /ine 

( ) f^^^i'"g on 

two parallel Jiraight 

» lines ( ■mmmim^mm and 

•), makes the alternate 



angles equal to one another ; and 
alfo tlie external equal to tlie in- 
ternal and oppojite angle on the 
fame Jide ; and the two internal 
angles on the fa?ne Jide together 
equal to two right angles. 



For if the alternate angles 



and 



▲ 



be not equal, 



draw 



», making 



A 



Therefore 



(pr- 23)- 
(pr. 27.) and there- 
fore two ftraight lines which interfed: are parallel to the 
fame flraight line, which is impoflible (ax. 1 2). 



Hence the alternate angles ^^ and ^|^ are not 
unequal, that is, they are equal: =: ^^^ (pr. 15); 

.*. jl^ = l/^ , the external angle equal to the inter- 
nal and oppofite on the fame iide : if ^^W be added to 



both, then 



A 



+ 




i 



^CLi 



(pr. 13)- 

That is to fay, the two internal angles at the fame fide of 
the cutting line are equal to two right angles. 

Q. E. D. 



BOOK I. PROP. XXX. THEOR. 



3^ 




TRAIGHT /mes ( _Z) 

lohich are parallel to the 

fame Jlratght line ( ), 



are parallel to one another. 




Let 



interfedl 



Then, 



= ^^ = iJB (pr. 29.), 



(pr. 27.) 



Q. E. D. 



32 BOOK I. PROP. XXXI. PROB. 




ROM a given 

point /^ to 
draw ajiraight 
line parallel to a given 
Jlraight line (——•). 




Draw 



from the point / to any point 



in 



make 
then — 




(pr. 23.), 
- (pr. 27.). 



Q. E. D. 



4 



BOOK I. PROP. XXXII. THEOR. 



33 




F any fide (- 



•) 



of a triangle be pro- 
duced, the external 

^figl^ ( ^^^) '-^ ^qual 
to the fum of the two internal and 




oppofte angles ( aiid ^^^ ) , 

and the three internal angles of 
every triangle taken together are 
equal to two right angles. 




Through the point / draw 
II (pr. 3i-)- 



Then 








(pr. 29.), 



and therefore 





(pr. 13.). 




J 



-dy 



Q. E. D. 



34 



BOOK I. PROP. XXXIII. THEOR. 




TRAIGHT fines (- 



and ) which join 

the adjacent extremities of 
two equal and parallel Jiraight 
~— — and "•»..---=. ), are 



themf elves equal and parallel. 



Draw 



the diagonal. 
(hyp.) 





and 



(pr. 29.) 



common to the two triangles ; 



■, and 



▼ = 4 



(pr. 4.) ; 



and /. 



(pr. 27.). 



Q. E. D. 



BOOK I. PROP. XXXIV. THEOR. 



35 




HE ofpofite Jides and angles of 
any parallelogram are equal, 
and the diagonal (i^— ^^— ) 



divides it into two equal parts. 



Since 




= A 
^ = t 



(pr. 29.) 




and 



common to the two triangles. 



/. \ 





\ (pr- 26.) 



and ^^W = ^^M (^^'^ ' 
Therefore the oppofite fides and angles of the parallelo- 
gram are equal : and as the triangles 



.N.""^ 



are equal in every refpect (pr. 4,), the diagonal divides 

the parallelogram into two equal parts. 

Q. E. D. 



36 BOOK I. PROP. XXXV. THEOR. 






ARALLELOGRAMS 

on the fame bafe, and 
between the fame paral- 
lels, are {in area) equal. 



and 



But, 



On account of the parallels, 

_Kpr. 29.) 
(Pi-- 34-) 

(pr. 8.) 



r=? 






minus 



minus 




r= 







Q^E. D. 



BOOK I. PROP. XXXVI. THEOR. 



37 




ARALLELO- 
GRAMS 



1 



is* 



( ^^ and ) on 

equal bafes, and between the 
fame parallels, are equal. 




Draw 



and ---..-— , 

■, by (pr. 34, and hyp.); 

= and II "— (pr. 33.) 



And therefore 



but 



J 
1.1 



is a parallelogram : 



(pr- 35-) 



(ax. I.). 



Q. E. D. 



38 BOOK I. PROP. XXXFII. THEOR. 




RIANGLES 



on the fame bafe (• 



■) 



and between the fame paral- 
lels are equal. 



Draw 



Produce 



\ fpr. ^i 



(pr- 3I-) 



1—M. and ^^ 



are parallelograms 
on the fame bafe, and between the fame parallels, 
and therefore equal, (pr. 35.) 



T 



=: twice 



f 



^ twice 



4 



(■ (pr- 34-) 



k.i 



Q. E D. 



BOOK I. PROP. XXXVIII. THEOR. 39 




RIANGLES 



;4H ^'ij JH 



(^Hi tind jm^ ) on 
equal bajes and between •■• 
the fame parallels are equal. 




Draw 
and 



II 



(pr. 31.). 



I # 



(pr. 36.); 



and 



■ i 
= twice ^^k 

^^ = twice ^H 



(pr- 34-)' 



i k 



(ax. 7.). 



Q^E. D. 



40 



BOOK I. PROP. XXXIX. THEOR, 





QUAL triangles 




\ 



and "^ on the fame bafe 
( ) and on the fame fide of it, are 

between the fvne parallels. 



If-^— ■», which joins the vertices 
of the triangles, be not || , 

draw II (pr.3i-). 



meeting 



Draw 



Becaufe 



(conft.) 



but 



W.4 



(pr- 37-) ■• 

(hyp.) ; 



A=4 



, a part equal to the whole, 
which is abfurd. 
Ji. ^i^-^-^ ; and in the fame 
manner it can be demonflrated, that no other line except 

is II ; .-. II . 

Q. E. D. 



BOOK I. PROP. XL. THEOR. 



41 




QUAL trian- 



gles 



( 



and M. 



) 




on equal bafes, and on the 
fame Jide, are between the 
fame parallels. 



If ■ which joins the vertices of triangles 
be not II - , 

draw — — — . II — -~— 



(pr. 31.), 



meeting 



Draw 



Becaufe 



(conft.) 



. ^^^- ^ 1^^^ , a part equal to the whole, 
which is abfurd. 

' 41" ~^^^"^ • ^"f^ in the fame manner it 
can be demonftrated, that no other line except 
— is II : .-. 



Q^E. D. 



42 



BOOK I. PROP. XLI. THEOR. 




Draw 



Then 





F a paral- 
lelogram 



A 




and a triangle ^^^ are upon 
the fame bafe — ^^^ and be tine en 
the fame parallels -.—---- and 
■ , the parallelogram is double 

the triangle. 



the diagonal ; 



V=J 



zz twice 




(pr- 37-) 



(pr- 34-) 



^^ 4 

^1^. ^ twice ^H^ . 



Q. E. D. 



BOOK I. PROP. XLII. THEOR. 43 




O conJiruSl a 
parallelogram 
equal to a given 



4 



triangle ^^/^andhaV" 
ing an angle equal to a given 

rectilinear angle , 



Make — — ^ = ■— « (pr. 10.) 
Draw , 

Draw I " [j ~' | (pr. 31.) 

^1^ := twice y 




(pr. 41.) 



but ^ z= lA (pr. 38.) 



4 



Q. E. D. 



44 BOOK I. PROP. XLIII. THEOR. 




HE complements 




and ^^^ cf 

the parallelograms ivhicli are about 
the diagonal of a parallelogram are 
equal. 





(pr- 34-) 



4. ^^ 

and JBL = ^ 



(pi-- 34-) 





(ax. 3.) 

Q. E. D. 



BOOK I. PROP. XLIV. PROB. 



45 



O a given 
Jlraight line 

ply a parallelo- 
gram equal to a given tri- 
angle ( ^^^^' ), and 





having an angle equal to 
a given reSiilinear angle 

( )■ 




g 




wi 



th 



▲ 



= ._i 



Make 

(pr. 42.) 

and having one of its fides -— — - conterminous 

with and in continuation of 1 m . 

Produce w^^mmm^ till it meets ' '■ " ■' || »»»■«»■ 

draw prnHnrp it fill if mpptg •■»■-,• continued ; 



draw •••««-.• II — «■■ meeting 

produced, and produce >•»■•»« 



but 





(pr. 430 



(conft.) 



▲ = ▼=▲ 



(pr. 19. and confl.) 
Q. E. D. 




BOOK I. PROP. XLF. PROP. 




O conjlrudl a parallelogram equal 
to a given reSlilinear figure 



( 



► 



) and having an 



angle equal to a given reSlilinear angle 



Draw 



and 



K.(t^m 



dividing 





to 



the redtilinear figure into triangles. 

Conftrudl 
having .„ 
— apply 



(pr.42.) 



having 




to 



having 



(pr. 44.) 

apply M = 

(pr. 44.) 



#=► 



##= >, 




and 



Mf mg is a parallelogram, (prs. 29, 14, 30.) 
having 



Q. E. D. 



BOOK I. PROP. XLVI. PROB. 



47 




PON a given Jlraight line 
(— ^^^) to conJlruB a 
fquare. 



Draw 

Draw • 
ing . 



and 



(pr. 1 1, and 3.) 



II 

drawn || 



>, and meet- 




In 



^ 



(conft.) 




=: a right angle (conft.) 

^H = Hp = ^ "g'^^ ^"gle (pr. 29.), 
and the remaining fides and angles muft 
be equal, (pr. 34.) 



and ,*, 




is a fquare. (def. 27.) 

Q. E. D. 



48 BOOK I. PROP. XLVII. THEOR. 





N a right angled triangle 

the fquare on the 
liypotenufe <• •< is equal to 



the fum of the fquares ofthejides, (■ 
and ). 



On 



and 



defcribe fquares, (pr. 46.) 



Draw -.—I 
alfo draw 



- (pr. 31-) 



and 





To each add 



T 



and 





Again, becaufe 



BOOK I. PROP. XLVII. THEOR. 



49 





and 




twice 



= twice ^H • 






In the fame manner it may be fhown 
that ^^ ^ 



hence 



## 




Q E. D. 




H 



so 



BOOK I. PROP. XLVIIL THEOR. 



/ 




F t/ie fquare 
of one Jide 

{ \ ) f 

a triangle is 

equal to the fquares of the 

other tivo fides (nn. i i 

and ), the angle 




( 



)fubtended by that 



fide is a right angle. 



Draw ■- 



and ^ 



(prs.11.3.) 



and draw —»-«--— alfo. 



Since 



(conft.) 



... "- + 

but ^ + - 

and — ^— i^- -|- 



+ 



(pr. 47-). 
- (hyp.) 



and ,*, 



confequently 





(pr. 8.), 



is a right angle. 



Q. E. D. 



51 




BOOK II. 



DEFINITION I. 




RECTANGLE or a 

right angled parallelo- 
gram is faid to be con- 
tained by any two of its adjacent 
or conterminous fides. 




Thus : the right angled parallelogram HH[ 
be contained by the fides — — — ^ and — 
or it may be briefly defignated by 



is faid to 



If the adjacent fides are eq^ual ; i. e. -— — — ^ ^ 

then — i^»^-« . - which is the expreflion 



for the redtangle under 



is a fquare, and 



is equal to J 



and 

- or 

- or 



52 



BOOK II. DEFINITIONS. 



DEFINITION II. 





N a parallelogram, 
the figure compokd 
of one ot the paral- 
lelograms about the diagonal, 
together with the two comple- 
ments, is called a Gnomon. 



Thus 




and 




are 



called Gnomons. 



BOOK II. PROP. I. PROP. 



53 




HE 7-e£langle contained 
by two ftraight lines, 
one of which is divided 
into any number of parts. 



= <;+ — 

/; equal to the fum of the reBangks 

contained by the undivided line, and the fever al parts of the 

divided line. 



I — — J— — i; 



Draw 



_L —— — and r= 



(prs.2.3.B.i.); 



complete the parallelograms, that is to fay, 



Draw 



\ (pr. 31- B.I.) 



L 



I 






+ 



- + 

Q. E. D. 



54 



BOOK II. PROP. II. THEOR. 




I 

I 



F a Jlraight line be divided 
into any tivo parts ' i , 

the fquare of the -whole line 
is equal to the fum of the 

reSlangles contained by the whole line and 

each of its parts. 




-f 



I 



Defcribe ■■-^^ (B. i. pr. 46.) 
Draw — parallel to ----- (B. i. pr. 31 ) 





I 



+ 



Q. E. D. 



BOOK 11. PROP. III. THEOR. 



55 



F a Jiraig/it line be di- 
vided into any two parts 
■ 11 ' , the reBangle 
contained by the "whole 
line and either of its parts, is equal to 
the fquare of that part, together with 
the reSf angle under the parts. 






m 

i 



= — ^ + 



or. 



Defcribe 



Complete 




I 



(pr. 46, B. I.) 



(pr. 31, B. I.) 



Then 




+ 



, but 



and 






In a fimilar manner it may be readily fhown 
that — . — zr m^'i _^ ——. — . 

Q. E. D 



56 



BOOK II. PROP. IF. THEOR. 




F a Jiraight line be divided 
into any tico parts , 

the fquare of the ii'hole line 
is equal to the fquare s of the 

parts, together ii-ith twice the reef angle 

contained by the parts. 




+ 



+ 



twice 



Defcribe 
draw - 



and 



4- 



vpr. 46, B. 1.) 
■ port. I.). 



(pr. 31, B. I.) 




4.4 



(pr. 5, B. I.), 



(pr. 29, B. I.) 




4 



500a: //. PROP. IF. THEOR. 57 



B 



/. by (prs.6,29, 34. B. I.) t,^J is a fquarc ^ — i 
For the fame reafons r I is a Iquare := ~"", 

« ""~ (pr, 43, b. I.) 



I 




b"t E— i = C-J+ — +— + 



B. 



twice >' • ■— , 

Q. E. D. 



58 



BOOK 11. PROP. V. PROP. 




F a Jlraight 
line be divided 



into two equal 
parts and alfo ^ 

into two unequal parts, 
the reSlangle contained by 
the unequal parts, together with the fquare of the line between 
the points of fe 51 ion, is equal to the fquare of half that line 



+ 




Defcribe IIHIH (pr. 46, B. i.), draw 
^ — II — -- 



and 



) 



II 



(pr.3i,B.i.) 



(p. 36, B. I.) 
■ - H (p. 43. B. I.) 



(ax. 2.) 



I- 



BOOK II. PROP. r. THEOR. 



59 



but 




and 




- (cor. pr. 4. B. 2.) 



(conft.) 



/. (ax. 2.) 



ifl.F- 



+ 



Q. E. D. 



6o 



BOOK II. PROP. VI. THEOR. 





F a Jlraight line be 
bifeSled ■ 

and produced to any 

point —^wmmmt , 

the reSlangle contained by the 
whole line fo increafed, and the 
part produced, together with the 
fquare of half the line, is equal 
to the fquare of the line made up 
of the half, and the produced part . 



Defcribe 




(pr. 46, B. I.), draw 
II 



and 



(pr. 3i,B. 1.) 




(prs. 36, 43, B. I ) 




but ^H = 



(cor. 4, B. 2.) 





+ 



(conft.ax.2.) 



Q. E. D. 




BOOK 11. PROP. VII. THEOR. 

F a Jlraight line be divided 
into any two parts wbmw^— , 
the fq liar es of the whole line 
and one of the farts are 
equal to twice the rectangle contained by 
the whole line and that part, together 
•with the fquare of the other parts. 



6i 



Defcribe 
Draw - 





and 



■ ■^■■■■« 



, (pr. 46, B. I.)- 
(poft. I.), 

(pr. 31, B. !.)• 



— I (pr- 43. -B. I.), 
add ■ = ■-' to both, (cor. 4, B- 2.) 



I 



(cor. 4, B. 2.) 



I 



+ ■ + 



+ 



■' + — ^ = 




+ 

Q. E. D. 



62 



BOOK II. PROP. VIII. THEOR. 



E3 




F ajlraight line be divided 

Into any two parts 

, the fquare of 

thefum of the whole line 

and any one of Its parts. Is equal to 

four times the reSlangle contained by 

the whole line, and that part together 

with the fquare of the other part. 






— + 



Produce 



and make 



Conftrudl 
draw 



J (pr. 46, B. 1.); 




(pr. 31, B. I.) 



but ^ + 



(pr. 4, B. II.) 

-^ z= 2. — 

(pr. 7, B. II .) 






•-+ — ^ 

+ °-' 

Q. E. D. 




BOOK 11. PROP. IX. THEOR. 

F a Jlraight 
line be divided 
into two equal 
parts ^— — ,.j y 



63 



and alfo into two unequal 

parts ^mmm^'^^m— 



^ the 

fquares of the unequal 

parts are together double 

the fquares of half the line, 

and of the part between the points offedlion. 

^ + ^= 2 ^ + 2 




Make — ■ _L and r= — 

Draw "..-.—«— and 

— II ,— II 



or 



and draw 




= 4 

4. = ^ 



(pr. 5, B.I.) ^ half a right angle, 
(cor. pr. 32, B. i.) 

(pr. 5, B. I.) =: half a right angle, 
(cor. pr. 32, B. i.) 

^ a right angle. 




4^ 



lence 



(prs. 5, 29, B. I.). 

wmmimtm^m^ ■■■■■* 

(prs. 6, 34, B. I.) 



+ 



^or + 



I 
I ■ 

+ 



\ 



(pr. 47, B. I.) 



+ 2 

Q. E. D. 



64 



BOOK II. PROP. X. THEOR. 





F a Jlraight line 
■ be bi- 
feBed and pro- 
duced to any point 
• — , thefquaresofthe 
•whole produced line, and of 
the produced part, are toge- 
ther double of the fquares of 
the half litie, and of the line 
made up of the half and pro- 
duced part. 



+ 



+ ^ 



Make 



and 



■— J_ and =1 to 
draw ^MvatMit and 



or 

- f 



(pr. 31, B. I.); 



draw 



alfo. 





4 



(pr. 5, B. I.) = half a right angle, 
(cor. pr. 32, B. i .) 

(pr. 5, B. I.) = half a right angle 
(cor. pr. 32, B. i.) 



4. 



m a right angle. 




BOOK II. PROP. X. THEOR. 6^ 

half a right angle (prs. 5, 32, 29, 34, B. i.), 




and 



.-.., (prs. 6, 34, B. I.). Hence by (pr. 47, B. i.) 

Q. E. D. 



66 



BOOK II. PROP. XI. PROP. 





O divide a given fir aight line -^^■■» 
in fuch a manner, that the reB angle 
contained by the whole line and one 
of its parts may be equal to the 



fquare of the other. 



Defcribe 

make «««■ 



1 1 ■ • «*»■ a 



n 



draw 



take 



on 



defcribe 



(pr. 46, B, I.), 
- (pr. 10, B. I.), 



(pr. 3, B. I.), 




(pr. 46, B. I.), 



Produce 



— (poft. 2.). 



Then, (pr. 6, B. 2.) 
2 _ i 



+ 



• •■■••■■ 



■"■', or, 





I 



Q^E. D. 



BOOK II. PROP. XII THEOR. 



67 




N any obtufe angled 
triangle, thefquare 
of the fide fubtend- 
ing the obtufe angle 
exceeds the fiim of the fquares 
of the fides containing the ob- 
tufe angle, by twice the rec- 
tangle contained by either of 
thefe fides andthe produced parts 
of the fa?ne from the obtufe 
angle to the perpendicular let 
fall on it from the oppofite acute 
angle. 




+ 



'' by 



^ + 



2 • 



+ 



By pr. 4, B. 2. 
^ + > + 2 

add — — — ^ to both 
2 _ V 



(pr. 47, B. I.) 



+ 



+ 



• or 



■ ; hence ' 
by 2 



'^ (pr. 47, B. I.). Therefore, 

^ • ' -"■ + ' + 



Q. E. D. 



68 



BOOK II. PROP. XIII. THEOR. 



FIRST. 



SECOND. 




^m 


F^ 




p^ 


Br^/^ 


^ 



N any tri- 
angle, the 
fquareofthe 
Jidefubtend- 
ing an acute angle, is 
lefs than the fum of the 
fquares of the Jides con- 
taining that angle, by twice the reSlangle contained by either 
of thefe fides, and the part of it intercepted between the foot of 
the perpendicular let fall on it from the oppofte angle, and the 
angular point of the acute angle. 



FIRST. 
+ ■ * by 2 



SECOND. 
.' -I *by 2 



+ 



2 • 



Firft, fuppofe the perpendicular to fall within the 

triangle, then (pr. 7, B. 2.) 
^■■■> ° -|- ^^^— ^ ^ 2 • ^^i^^"»« • — — -^ ■■■• 

add to each ^ihi^'^ then, 
I..... "■ -|- _• ^4- - = 2 • ■— • - 

+ ' + « 

/. (pr- 47. B. I.) 



+ 



BOOK 11. PROP. XIII. THEOR. 69 

and .*. ^ Z] ^— "— - + — - by 

2 • -■" • ■■■-■i™ . 

Next fuppofe the perpendicular to fall without the 
triangle, then (pr. 7, B. 2.) 

add to each — ■— ■ - then 

+ ^ + 2 ... (pr. 47, B. I.), 

■J 1 <2 ^_ „ I a 

1^— -|- -^.— ^ 2 • ^mM»» . _l-_ -J- ', 

Q. E. D. 



7° 



BOO A' //. PROP. XIV. PROB. 




O draw a right line of 
•which the fquare flmll be 
equal to a given reSli- 
linear figure . 



fuch that. 






Make ^^^^H = ^^V (pr. 45, B. i.), 

produce "•- until — — -■. := • 

take -■■■.«—- ^ i^— — (pr. 10, B. i.), 

Defcribe f \ (poft. 3.), 

and produce -^^— to meet it : draw — — ^— , 

(pr. 5, B. 2.), 
but — ■ = ' ' " + — "— -(pr. 47, B. I.); 

• wmmm^t" ^I ■■■■■■ • «■»« , and 

Q. E. D. 



BOOK III. 




DEFINITIONS. 
I. 

QUAL circles are thofe whofe diameters are 
equal. 



II. 



A right line is said to touch a circle 
when it meets the circle, and being 
produced does not cut it. 



III. 

Circles are faid to touch one an- 
other which meet but do not cut 
one another. 




IV. 

Right lines are faid to be equally 
diftant from the centre of a circle 
when the perpendiculars drawn to 
them from the centre are equal. 




72 



DEFINITIONS. 




And the ftraight line on which the greater perpendi- 
cular falls is faid to be farther from the centre. 



VI. 

A fegment of a circle is the figure contained 
by a ftraight line and the part of the circum- 
ference it cuts off. 

VII. 

An angle in a fegment is the angle con- 
tained by two ftraight lines drawn from any 
point in the circumference of the fegment 
to the extremities of the ftraight line which 
is the bafe of the fegment. 

VIII. 

An angle is faid to ftand on the part of 

; the circumference, or the arch, intercepted 

between the right lines that contain the angle. 

IX. 

A fed:or of a circle is the figure contained 
by two radii and the arch between them. 





DEFINITIONS. 



11 



Similar fegments of circles 
are thofe which contain 
equal angles. 




Circles which have the fame centre are 
called concentric circles. 




74 



BOOK III. PROP. I. PROB. 




O Jind the centre of a given 
circle 



o 



Draw within the circle any ftraight 



Hne — ^ 

draw 
hi left . 



ma 



ke. 



i^MMMi ^ and the point of 
biledtion is the centre. 

For, if it be pofTible, let any other 
point as the point of concourfe of .^— — , ---..--- 
and — .— — be the centre. 



Becaufe in 





and 



■ ----— (J'^yp- ^""^ 2* I J def. 15.) 
-- (conft.) and ••■- common, 

^B. I, pr. 8.), and are therefore right 



angles ; but 



^ = ^_| (con 



ft. 



(ax. I I .) 



which is abfurd ; therefore the aflumed point is not the 
centre of the circle ; and in the fame manner it can be 
proved that no other point which is not on — ^^— • is 
the centre, therefore the centre is in ^— ^^— , and 
therefore the point where 1 is bifedled is the 

centre. 

Q. E. D. 



BOOK III. PROP. 11. THEOR. 



75 




STRAIGHT line C—) 
joining two points in the 
circumference of a circle 



lies ivholly within the circle. 



Find the centre of 



o 



(B.S-pr.i.); 




from the centre draw 



to any point in 



meeting the circumference from the centre ; 
draw — — — and . 



Then 



= -^ (B. i.pr. 5.) 



but 



or 



CZ ^ (B. I. pr. 16.) 
(B. I. pr. 19.) 



but 



.*. every point in 



lies within the circle. 
Q. E. D. 



76 



BOOK III. PROP. III. THEOR. 




Draw 



F a Jlraight line ( 
drawn through the centre of a 



circle 



o 



bife£lsachord 



( •'•■) which does not paj's through 

the centre, it is perpendicular to it; or, 
if perpendicular to it, it bifeSls it. 



and 



to the centre of the circle. 



In >^ I and L..._V 



■• •■ ■■» 



and ,*, 



m^^^ common, and 

= (B. 1. pr. 8.) 

_L -..«.- (B. I. def. 7.) 




Again let 
Then in 



J and L^„..T^ 




(B. i.pr. 5.) 

(hyp.) 



and 



and .*. 



(B. I. pr. 26.) 



bifedts 



Q. E. D. 



BOOK HI. PROP. IF. THEOR. 



11 




F in a circle tiaojlraight lines 
cut one another, which do 
not both pafs through the 
centre, they do not bifeSl one 



another. 



If one of the lines pafs through the 
centre, it is evident that it cannot be 
bifecfted by the other, which does not 
pafs through the centre. 




But if neither of the Hnes — =— ^^— or •— ^-i— 
pafs through the centre, draw ——----. 
from the centre to their interfedlion. 

If «i^^^ be bileded, ._._._ _L to it (6. 3. pr. 3.) 
.*. ^^ = I ^ and if — be 



bifed:ed, 



(B. 3- P'-- 3-) 



and .*, 






5 a part 



equal to the whole, which is abfurd : 
.*. — —— — and — — — • 



do not biiecfl one another. 



Q. E. D. 



w* 



78 



BOOK III. PROP. V. THEOR. 




F two circles 
interfeSl, they have not the 



(0) 



Janie centre. 



Suppofe it poflible that two interfedting circles have a 
common centre ; from fiich fuppofed centre draw ^.i^.. 
to the interfering point, and ^^—^— ••■--■■ 
the circumferences of the circles. 



meetmg 



Then 
and <— 



(B. i.def 15.) 
- (B. I. def. 15.) 
«»- J a part 



equal to the whole, which is abfurd : 

.', circles fuppofed to interfedl in any point cannot 

have the fame centre. 

Q,E. D. 



BOOK III. PROP. VL THEOR. 



79 




F tivo circles 




touch 



one another internally, they 



have not the fame ce?itre. 




For, if it be poffible, let both circles have the fame 
centre; from fuch a fuppofed centre draw ---■» 
cutting both circles, and ■— — ^— to the point of contadl. 



Then 
and — 



(B. i.def. 15.) 
(B. I.def. 15.) 
J a part 



equal to the whole, which is abfurd ; 
therefore the aiTumed point is not the centre of both cir- 
cles ; and in the fame manner it can be demonftrated that 
no other point is. 

g E. D. 



8o 



BOOK HI. PROP. FII. THEOR. 



nCURE 1. 




FIGURE II. 




F Jt'om any point within a circle 



which is not the centre, lines 



o 




are drawn to the circumference ; the greatejl of thofe 
lines is that (-i^.«i"») which pajfes through the centre, 
and the leaf is the remaining part ( ^ of the 

diameter. 

Of the others, that ( ^— ■— — > ) which is nearer to 
the line pafing through the centre, is greater than that 
( mmmmm^^ ) wliich Is itiore remote. 



Fig. 2. The two lines (• 



and 



) 



which make equal angles with that pafpng through the 
centre, on oppofite fdes of it, are equal to each other; and 
there cannot be drawn a third line equal to them, from 
the fame point to the circumference. 



FIGURE I. 

To the centre of the circle draw —-—— and -— «-- — j 
then "— -— rr —..—.. (B. i. def. 15.) 
......i^Mi ^ ^^— -|- ...«>. C — — ^-» (B.I . pr. 20.) 

in like manner — — — (• may be fhewn to be greater than 
.i.M__- , or any other line drawn from the fame point 



to the circumference. Again, by (B. i. pr. 20.) 

take — — from both ; /. — — — CI ....1^— (ax.), 
and in like manner it may be fhewn that — ^— ^ is lefs 



BOOK III. PROP. VII. THEOR. 8i 

thiin any other line drawn from the fame point to the cir- 



cumference. Again, in y*/ and 




common, ^^ [^ IV , and 



(B. I. pr. 24.) and 



may in like manner be proved greater than any other line 
drawn from the fame point to the circumference more 
remote from -——■-, 



FIGURE II. 

If ^=^. hen = ,if„o. 

take — — ^— r= ^— — draw '■"■'■, then 

^c— 'I :>»^ 

in ^^ I and I ,^^ , — — common. 






(B. i.pr. 4.) 



a part equal to the whole, which is abfurd : 
■"— — IS * and no other line is equal to 

■^ drawn from the fame point to the circumfer- 



ence ; for if it were nearer to the one pafling through the 
centre it would be greater, and if it were more remote it 
would be lefs. 

Q. E. D. 



M 



82 



nOOK HI. PROP. nil. THEOR. 



The original text of this propolition is here divided into 
three parts. 




I. 



^Voll ^^ f''°'" '' P°'"^ without a circle, Jlraight 

iiKCs \ — — — \ ore (jrd'wn to 



V»y 



the cir- 



cu/nference ; of thofe falling upon the concave circum- 
ference the greatejl is that ( — ») ichich fafja 
through the centre, and the line ( i ) ichich is 
nearer the greatejl is greater than that ( ) 
'ichich is more remote. 



Draw 



and .■■-..>... to the centre. 



Then. ■— . which pallcs through the centre, is 

greatcll; for fince — — nz , it ^-^-^— 

be added to both. -•■-■ ^ — ^— -|- ; 

l^iit C — — — {^- 1- P'- -^•) •'- — — — is greater 
than ;inv other line dr.iwn from the fame point to the 
concave circumference. 



Again in 



and 



BOOK rrr. prop. nir. tiikor 

and — ^— ^ common, Init 







(B. I. pr. 24.); 



and in like manner 



may be Ihcwn ZZ t'l-"! -^'7 



other line more remote from 



II. 

Of thofc lines falling on the convex circumference the 
leaf is that (————) which being produced would 
pafs through the centre, and the line which is nearer to 
the lea/l is Icf than that which is more remote. 



For, fince — — — -j- 
and 



/« 




And again, fince — — -|- 

h (B. i.pr. 21.), 

and — — rs . 



— — -, And lb of others. 



III. 



Alfo the lines making equal angles with that which 
pajjes through the centre arc equal, whether /ailing on 
the concave or convex circumfrence ; and no third line 
can he drinvn equal to the/n from the f<imc point to the 
circumference. 



Forif •■ 
make 



•— ^ C "■"•"", Init making ^ =: ^ ; 

----- ^ ----- ^ ;iml tl|;l\V ...... -.^ 




84 



BOOK III. PROP. Fin. THEOR. 



Then in 



and 



) and / 



; 



/ 



we have 



common, and alfo ^ =: 41, 
- = (B. I. pr. 4.); 



but 



which is abfurd. 



of 



■ ■■iisBisB IS not Z!Z 
----- — 9 .*. --■■ 



-, nor to any part 

is not r~ — -----^ 



Neither is 



'— , they are 



to each other. 



And any other line drawn from the fame point to the 
circumference muft He at the fame fide with one of thefe 
lines, and be more or lefs remote than it from the line pall- 
ing through the centre, and cannot therefore be equal to it. 



Q. E. D. 



BOOK in. PROP. IX. THEOR. 



85 




F a point be taken ivithin a 
circle ( ] , from which 



o- 



more thwi two equal Jlraight lines 

can be drawn to the circumference^ that 
point tnuji be the ceiitre of the circle. 

For, if it be fuppofed that the point |^ 
in which more than two equal ftraight 
lines meet is not the centre, fome other 
point — .. muft be; join thefe two points by 
and produce it both ways to the circumference. 



Then fince more than two equal ftraight lines are drawn 
from a point which is not the centre, to the circumference, 
two of them at leall; muft lie at the fame fide of the diameter 




.; and fince from a point 



/\ 



w 



hich 



is 



not the centre, ftraight lines are drawn to the circumference ; 

the greateft is -i^— .--= = ^ which pafies through the centre : 

and — ^— — — which is nearer to ^^■—••«'. |^ ———~- 

which is more remote (B. 3. pr. 8.) ; 

but — — — ^— rr ^— ^— =- (hyp.) which is abfurd. 

The fame may be demonftrated of any other point, dif- 
ferent from / |\^ which muft be the centre of the circle, 

Q. E. D. 



86 



BOOK III. PROP. X. THEOR. 




NE circle I } cannot interfeSl another 
I J /« more points than two. 



For, if it be poffible, let it interfedl in three points ; 
from the centre of ( I draw 



O 



to the points of interfedlion ; 



(B. I. def. 15.), 

but as the circles interfedl, they have not the fame 
centre (B. 3. pr. 5.) : 



.*, the alTumed point is not the centre of 



o. 



and 



and 



are drawn 



from a point not the centre, they are not equal (B. 3. 
prs. 7, 8) ; but it was fhewn before that they were equal, 
which is abfurd ; the circles therefore do not interfedl in 
three points. 

Q. E. D. 



BOOK III. PROP. XL THEOR. 



87 




F two circles 



o 



o 



and 



touch one another 



internally, the right line joining their 
centres, being produced, Jliall pafs through 
a point of contaSl. 



For, if it be poffible, let 




join their centres, and produce it both 
ways ; from a point of contadl draw 

— — — to the centre of ( J , and from the fame point 
of contadl draw -•-•--— to the centre of I J . 



Becaufe in 



4 



+ 



■■■•■•■•t 



(B. I . pr. 20.), 



and 



o 



as they are radii of 



8B BOOK III. PROP. XI. THEOR. 

but — ^ -|- I rr — — — ; take 

away — — — which is common, 

hut — i^— i ^ -- — --^ 



ii of r^ , 



becaufe they are radi 

and ,*, --»-" C ^^ ^ P^i't greater than the 

whole, which is abfurd. 

Tlie centres are not therefore fo placed, that a line 
joining them can pafs through any point but a point of 
contadt. 

Q. E. D. 



BOOK III. PROP. XII. THEOR. 



89 




F two circles 



o 



t/ier externally, the Jiraight line 
1 1 joining their centres, 

pajfes through the point of contaB. 




touch one a7io 



If it be polTible, let 



join the centres, and 



not pafs through a point of contadl ; then from a point of 
contad: draw -"^^^== and "" ■ ••^■^- ' '-• to the centres. 



Becaufe 

and . 
and • 



. + 



(B. I. pr. 20.), 

= (B. I. def. 15.), 

= ^ (B. I. def. 15.), 



+ 



', a part greater 



than the whole, which is abfurd. 



The centres are not therefore fo placed, that <-he line 
joining them can pafs through any point but the point of 
contadl. 



Q.E. D. 



90 BOOK in. PROP. XIIL THEOR. 



FIGURE I. 



FIGURE II. 




FIGURE III. 




NE circ/e can- 
not touch ano- 
ther, either 
externally or 

internally, in more points 

than one. 



Fig. I . For, if it be poffible, let 
and f 1 touch one 



o 



another internally in two points ; 
draw — — - joining their cen- 
tres, and produce it until it pafs 
through one of the points of contadl (B. 3. pr. 11.); 
draw — — — and •^— ^—^ , 



.-. if 



(B. I. def 15.), 



be added to both, 
+ 



but 
and .*. 



+ 



+ 

which is abfurd. 



.■ (B. I. def 15.), 

- = — _— ; but 
— — (B. I. pr. 20.), 



BOOK III. PROP. XIII. THEOR. 91 

Fig. 2. But if the points of contadl be the extremities 
of the right line joining the centres, this ftraight line muft 
be bifedled in two different points for the two centres; be- 
caufe it is the diameter of both circles, which is abfurd. 



, let f j and I J 



Fig. 3. Next, if it be poffible 



touch externally in two points; draw — — joining 

the centres of the circles, and paffing through one of the 
points of contact, and draw — i— — — and -^^— — , 



— -^ z= _ (B. I. def. 15.); 
nd ...«■■•. zr I (B. i. def. 15.); 



-\- — — -^ Z:Z. ■BMBsaaa * but 



+ ^-^™''— C ».-- (B. I. pr. 20.), 



which is abfurd. 

There is therefore no cafe in which two circles can 
touch one another in two points. 

Q E. D. 



92 



BOOK III. PROP. XIV. THEOR. 




Then 
and 



infcribed in a circle are e- 
qually dijiantfrom the centre ; 
andalfofjlraight lines equally 
dijiafit from the centre are equal. 




From the centre o 



o 



draw 



-L 



to — 
,join 



■••■■ and --• 

^— and — ■ 



fince 



= half '" (B. 3. pr. 3.) 

= i (B. 3-pr-3-) 

= — (hyp.) 



and 



(B. I. def. 15.) 



and 



but fince > - s^ is a right angle 

= ' + MB.i.pr.47-) 

.' = ' + ^ for the 



.% -^ + 



fame reafon. 



BOOK III. PROP. XIV. THEOR. 93 



Alfo, if the lines 1 . .■.•»■ and •— i»«...«.r be 

equally diflant from the centre ; that is to fay, if the per- 
pendiculars -■-■•■■■■■■ and -m........ be given equal, then 



For, as in the preceding cafe, 



. ^ :::= __i.^, and the doubles of thefe 
....... and ^n....... are alfo equal. 

Q. E. D. 



94 



BOOK III. PROP. XV. THEOR. 



FIGURE I. 




but 



HE diameter is the greatejl Jiraight 
line in a circle : and, of all others, 
that which is neareji to the centre is 
greater than the more remote. 



FIGURE I. 
The diameter ^^ is CZ any line 

For draw ' and •••••••••• 

Then .••■>••»■■> ^ ^— ^._i 
and •^— ^— = — — — . 



mXm ■■■■•»««» ^^ 



■■■•■■•■• 



(B. I . pr. 20.) 



Again, the Hne which is nearer the centre is greater 
than the one more remote. 

Firft, let the given lines be — — ^ and - , 

wnich are at the fame fide of the centre and do 
not interfed: ; 

draw J ' 



BOOK III. PROP. XF. THEOR. 



95 




FIGURE II. 
Let the given lines be ■^— ^ and ^i— ^ 
which either are at different fides of the centre, 
orinterfedt; from the centre draw - -■•-- 

and -»-«-—- J_ -^^Mi->i» and , 



FIGURE II. 



make 
draw 



Since 



and 



the centre, 
but — — — 



and 



are equally diflant from 
(B. 3. pr. 14.); 




(Pt. i.B. 3.pr. 15.), 



Q. E. D. 



96 



500 A' ///. PROP. XVI. THEOR. 




llEJiraiglit 
line - 

draii-n 
from the 
extremity of the diame- 




ter 



of a 



circle 



h 



perpendicular to it falls 
••.^^ ^,., without the circle. 

||» • '^ * And if anyjlraight 

*** line ........ be 

drawn from a point 
————— within that perpendi- 



cular to the point of contaB, it cuts the circle. 

PART I 

If it be poffible, let ^ which meets the circle 

again, be _L , and draw , 



Then, becaufe 



^ = ^ (B. i.pr. 5.), 
and .'. each of these angles is acute. (B. i. pr. 17.) 

but ^^ =r I J (hyp.), which is abfurd, therefore 

ii...._ drawn _L — ^^^— does not meet 
the circle again. 



BOOK in. PROP. XVI. THEOR. 97 

PART 11. 

Let — Bi*—"— be _L -^-^^— and let -— — - be 
drawn from a point y between — ■— ■— • and the 
circle, which, if it be poflible, does not cut the circle. 

Becaufe H^ =: | ^ , 

.*. ^^ is an acute angle ; fuppofe 
....... ...4.... _L .-■«•-•-, drawn from the centre of the 

circle, it mufl; fall at the fide of ^^ the acute angle. 
,*, B^^ which is fuppofed to be a right angle, is C ^^; 



but ............ = , 

and .*, --••••.. ^ ......«..■—, a part greater than 

the whole, which is abfurd. Therefore the point does 
not fail outfide the circle, and therefore the ftraight line 
■ ••••MiM* cuts the circle. 

Q. E. D. 



98 



BOOK III. PROP. XVII. THEOR. 





O Jraiv a tangent to a given 
circle \ \ from a 



o 



given point, either in or outjide of its 
'•♦^ circumference. 



If the given point be in the cir- 
cumference, as at I , it is plain that 

the ftraight line "■■" J_ -«— — - 
the radius, will be the required tan- 
gent (B. 3. pr. 16.) But if the given point ^ 
outlide of the circumference, draw — 

from it to the centre, cutting 



be 



draw •■««■■■■** ^_ 



concentric with 
then 



o 




.., defcribe 



radius^ .■■■■ub^, 
will be the tangent required. 



BOOK III. PROP. XFII. THEOR. 



99 






XV 



/ 



/ 



For 



in 



and i\. 



, ^^^ common, 




(B. I. pr. 4.) flB =: ^^^ ^ a right angle, 
.*. ^a^a^B is a tangent to 

Q. E. D. 



o 



loo BOOK III. PROP. XFIII. THEOR. 




and .*, 




F a right line •• — be 

a tangent to a circle, the 
fir aight line — ^— draivn 
from the centre to the 
i point of contaSl, is perpendicular to it. 



For, if it be poflible, 
let *>■ be ^ — ... 



then becaufe 



= [^ 




is acute (B. i . pr. 17.) 

c 



(B. I. pr. 19.); 



but 



»•*•■ , a part greater than 



the whole, which is ablurd. 



/, .»«.. is not _L -"•—•••5 and in the fame man- 

ner it can be demonftrated, that no other line except 
— ■— — is perpendicular to «•-.■...-• , 

Q. E. D. 



BOOK III PROP. XIX. THEOR. 



lOI 




F a Jlra'tght line 



be a tangent to a circle, 
thejiraight line , 

drawn perpendicular to it 

from point of the contact, pajfes through 

the centre of the circle. 

For, if it be pofTible, let the centre 



be without « 



and draw 



• ••- from the fuppofed centre 
to the point of contadl. 



Becaufe 



X 





(B. 3.pr. i8.) 

.'. ^^ =: I Ji , a right angle ; 

but ffj^ = I 1 (hyp.)' and /. ^ = 

a part equal to the whole, which is abfurd. 



Therefore the affumed point is not the centre ; and in 
the fame manner it can be demonftrated, that no other 
point without _„_^ is the centre. 



Q. E. D. 



102 



BOOK III. PROP. XX. rUEOR. 



FIGURE I 




HE angle at the centre of a circle, is double 
the angle at the circumference, when they 
have the fame part of the circumference for 
their bafe. 



FIGURE I. 
Let the centre of the circle be on 



a fide of ^ , 



Becaufe 



i = ^ 



= ^ (B. i.pr.5.). 



But 




+ 



\ 



or 




=: twice . (B. i. pr. 32). 



FIGURE II. 




FIGURE II. 

Let the centre be within ^ ^ the angle at the 
circumference ; draw ■■■^^^— from the angular 
point through the centre of the circle ; 

then ^ := r ? and = ^^ , 

becaufe of the equality of the fides (B. i. pr. 5). 



BOOK III. PROP. XX. THEOR. 103 



Hence 



-|- ^ -|~ "I" ^ twice 

But ^ = ^ + ^ , and 









r= twice 



FIGURE III. 
Let the centre be without W and 



FIGURE III. 





draw m^ 




the diameter. 


B( 


jcaufe ▼ 




: twice ^ ; i 




r= 


twice 


^^ (cafe I .) ; 




• 
• • 


A 


^ twice ▼ , 




Q. E. D. 



I04 BOOK III. PROP. XXI. THEOR. 



FIGURE I. 




HE angles ( ^^ , ^^ ) in the fame 
Jegment of a circle are equal. 



FIGURE I. 
Let the fegment be greater than a femicircle, and 
draw — ^^^— and — — — — to the centre. 




twice ^^ or twice 

(B. 3. pr. 20.),- 

4 = 4 



4 



FIGURE II. 




FIGURE II. 
Let the fegment be a femicircle, 01 lefs than a 
femicircle, draw —— — the diameter, alfo draw 



^=4a„dV = ^ 





(cafe I.) 



Q. E. D. 



J 



BOOK III. PROP. XXII. THEOR. 105 




HE oppojite arigJes 




Af 



and ^^ . ^^1 and 

^r of any quadrilateral figure in- 
fcr'ibed in a circle, are together equal to 
two right angles. 



Draw 



and 




the diagonals ; and becaufe angles in 

the fame fegment are equal ^W =: ^^ 
and ^r rr: ^^ | 
add ^ to both. 

two right angles (B. i. pr. 32.). In like manner it may 
be Ihown that, 

Q. E. D. 



io6 BOOK III. PROP. XXIII. THEOR. 





PON t/ie fame 
Jlraight line, 
and upon the 
fame fide of it, 
two fmilar fegments of cir- 
cles cannot be conflruBed 
which do not coincide. 



For if it be poflible, let two fimilar fegments 



o 



and 




be conftrudled ; 



draw any right line 
draw « 



• cutting both the fegments, 
and ^^-HMM . 



Becaufe the fegments are fimilar. 



(B. 3. def 10.), 






but ^M [Z ^^ (B. I. pr. 16.) 

which is abfurd : therefore no point in either of 

the fegments falls without the other, and 

therefore the fegments coincide. 

Q. E. D. 



BOOK III. PROP. XXIV. THEOR. 



107 




IMILAR 

fegments 



and 




, of cir- 



cles upon equal Jlraight 
lines ( '^^^ and — — ) 
are each equal to the other. 




For, if 
that — 



be fo applied to 



- may fall on ^— ^— 
may be on the extremities 



the extremities of 



and 



at the fame fide as 



becaufe 



muft wholly coincide with 



and the fimilar fegments being then upon the fame 

flraight line and at the fame fide of it, muft 

alfo coincide (B. 3. pr. 23.), and 

are therefore equal. 

Q. E. D. 



io8 



BOOK III. PROP. XXV. PROP. 




SEGMENT of a circle 
being given, to defcribe the 
circle of 'which it is the 
feginent. 



From any point in the fegment 
draw ^^— ^ and — ^^^^ bifeft 
them, and from the points of biledlion 



draw 
and 



where they meet is the centre of the circle. 



Becaufe — ..__ terminated in the circle is bifedied 
perpendicularly by ^■"■■"^ , it palTes through the 
centre (B. 3. pr. i.), likewife ^a^^M^ pafles through 
the centre, therefore the centre is in the interfedlion of 
thefe perpendiculars. 

Q.E. D. 



BOOK III. PROP. XXVI. THEOR. 109 




N egua/ circles 



the arcs 



O ""' o 



on which 



Jiand equal angles, whether at the 


centre or 


circum 


ference, are equal. 








Firfl, let ^^ 




at the 


centre. 


Then fince 


= 


mmm ^ 

O- 




/\ 


and ^♦;;.„ 


■•\ 


have 





■ ■■■■■■ 



and 



But 



▲ =▲ 



(B. i.pr.4.). 



(B.3.pr. 20.); 



• • O '"' o 



are fimilar (B. 3. def. 10.) ; 
they are alfo equal (B. 3. pr. 24.) 



110 BOOK III. PROP. XXVI. THEOR. 

If therefore the equal fegments be taken from the 
equal circles, the remaining fegments will be equal ; 



hence 



(ax. 3.); 



and .*, 



But if the given equal angles be at the circumference, 
it is evident that the angles at the centre, being double 
of thofe at the circumference, are alfo equal, and there- 
fore the arcs on which they fland are equal. 

Q. E. D. 



BOOK III. PROP. XXVn. THEOR. 1 1 1 




N equal circles. 



O-O 



the angles and ^^ which Jiand upon equal 

arches are equal, whether they be at the centres or at 
the circumferences. 




For if it be poffible, let one of them 



▲ 



be greater than the other 
and make 



▲ 



\ = 4 



/. V_^-" = **»n„..« (B. 3. pr. 26.) 

but V«^ = **♦.....•♦ (hyp.) 

.'. ^-i_ ^ := > fc^ _ ^ ^ ^ a part equal 

to the whole, which is abfurd ; .*, neither angle 

is greater than the other, and 

,*, they are equal. 

Q. E. D. 




••••.•■••• 



112 BOOK III. PROP. XXVIII. THEOR. 





N equa/ circles 



o-o 



egual chords 
arches. 



cut off equal 




From the centres of the equal circles. 



draw 



and 



and becaufe 



c=o 



alfo 



(IW-) 




(B. 3. pr. 26.) 



and 



.0 = 



(ax. 3.) 
Q. E. D. 



BOOK III. PROP. XXIX. THEOR. 113 




N equal circles 



nd ••-- ivhich fub~ \ ^ ^^ / 



the chords ^— -^^ and 
tend equal arcs are equal. 

If the equal arcs be femicircles the propofition is 
evident. But if not, 
let — ^^i^ . — — i^ , and 



be drawn to the centres ; 



becaufe 

and 
but ^— — ^ and 





(hyp.) 

(B.3.pr.27.); 

•■»...... and «•-' 

(B. I. pr. 4.); 




but thefe are the chords fubtending 
the equal arcs. 



Q. E. D. 



114 



BOOK III. PROP. XXX. PROB. 





O l>ife^ a given 



n- 



arc 



draw 



Draw 
make — ^ 
_L — ^^-■" , and it bifedls the arc. 



■*•«•■•■ 



Draw •"••■"»■ and 



and 



— --— (confl:.), 
is common, 

(conft.) 
. (B. i.pr.4.) 




= y"*'\ (B. 3- pr. 28.). 
and therefore the given arc is bifeded. 



Q. E. D. 



BOOK III. PROP. XXXI. THEOR. 115 




N a circle the angle in afemicircle is a right 
angle, the angle in a fegment greater than a 

femicircle is acute, and the angle in a feg- 
ment lefs than a femicircle is obtufe. 



FIGURE I. 



FIGURE I. 
The angle ^ in a femicircle is a right angle. 



V 




Draw 



and 



and 



V 



= ^ (B. i.pr. 5.) 



+ A= V 



^ the half of two 



right angles sz a right angle. (B. i. pr. 32.) 



FIGURE II. 
The angle ^^ in a fegment greater than a femi- 
circle is acute. 



FIGURE II. 



Draw 




the diameter, and .- 
^ a right angle 

^^ is acute. 




ii6 BOOK III. PROP. XXXI. THEOR. 



FIGURE III. 




FIGURE III. 
The angle ^^^^ in a fegment lefs than femi- 

circle is obtufe. 

Take in the oppofite circumference any point, to 
which draw «mmm* and . 



^ 




Becaufe W^ -|- 

(B. 3. pr. 22.) 



^Oh 



but 



(part 2.), 




is obtufe. 



Q. E. D. 



BOOK III. PROP. XXXIL THEOR. 117 




F a rig/it line ^■■■■ii"— ■ 
be a tangent to a circle, 
and frotn the point of con- 
tact a right line " 
be drawn cutting the circle, the angle 

jg^ made by this line with the tangent 

is equal to the angle ^^ in the alter- 
ate fegment of the circle. 




If the chord fhould pafs through the centre, it is evi- 
dent the angles are equal, for each of them is a right angle. 
(B. 3. prs. 16, 31.) 



But if not, dra'V 



from the 



point of contadl, it muft pafs through the centre of the 
circle, (B. 3. pr. 19.) 

.-. ^ = ^ (B.3.pr.3i.) 

W + f =• CA = f (B- I- pr. 32.) 
/. ^ = ^ (ax.). 

Again CJ = iV\ = _ + ^ 
(B. 3. pr. 22.) ^ 

/. C. y = ^m » (ax.), which is the angle in 

the alternate fegment. 

Q. E. D. 



ii8 BOOK III. PROP. XXXIII. PROP. 





N a given ftraight line ^^^— 
to dejcribe a Jegment of a 
circle that Jliall contain an 
angle equal to a given angle 



^,ty, 



If the given angle be a right angle, 
bifedt the given line, and defcribe a 
femicircle on it, this will evidently 
contain a right angle. (B. 3. pr. 31.) 



If the given angle be acute or ob- 
tufe, make with the given line, at its extremity. 



, draw 



make 
with 



f 



defcribe 




and 



or — ' ■ ■ ■ as radius, 
for they are equal. 



is a tangent to 



o 



(B. 3. pr. 16.) 



divides the circle into two fegments 



capable of containing angles equal to 
/ W and j/^ which were made refpedlively equal 



and 



(B. 3.pr. 32.) 



Q. E. D. 



BOOK III. PROP. XXXIV. PROP. 119 




O cut off from a given cir- 



cle 



o 



a fegment 



which fiall contain an angle equal to a 



given angle 




I>raw 



(B. 3. pr. 17.), 



a tangent to the circle at any point ; 
at the point of contad: make 






and 



> 



the given angle ; 
contains an angle ^ the given angle. 



Becaufe 
and « 



angle in 



> 



• IS a tangent, 
cuts it, the 

(B. 3. pr. 32.), 




but 





(conft.) 



Q. E. D. 



120 BOOK III. PROP. XXXV. THEOR. 



FIGURE I. 




FIGURE II. 





F two chords \ •• ••"" i .^ ^ circle 

interfeSl each other, the reBangle contained 
by the fegments of the one is equal to the 
re 51 angle contained by the fegments of the other. 



FIGURE I. 
If the given right lines pafs through the centre, they are 
bifedled in the point of interfed:ion, hence the recftangles 
under their fegments are the fquares of their halves, and 
are therefore equal. 



FIGURE II. 
Let — "— - pafs through the 'centre, and 

.«■>■.■- not; draw — — — — and . 



Then 



X 



or 



» (B. 2. pr. 6.), 

X = ' 



X - = 

■■ (B. 2. pr. 5.). 



X 



FIGURE III. 




FIGURE III. 
Let neither of the given lines pafs through the 
centre, draw through their interfedlion a diameter 
........ 9 

and X = X 

>■■■•■ (Part. 2.), 

alfo - X = X 

........ (Part. 2.) ; 

Q. E. D. 



BOOK III. PROP. XXXFI. THEOR. 



121 




F from a point without a FIGURE I. 

circle twojlraight lines be 

drawn to it, one of which 

— ■'^"» is a tangent to 
the circle, and the other — — -- 
cuts it ; the re^angle under the whole 
cutting line — ••«■• and the 

external fegment ^-^ is equal to 
the fquare of the tangent -^— , 

FIGURE I. 

Let — i— •• pafs through the centre; 

draw from the centre to the point of contadl ; 

- (B. i.pr. 47), 




minus 



or 



mmus 






(B.2.pr. 6). 



FIGURE II. 
If ■"•'■ do not 

pafs through the centre, draw 



FIGURE n. 



and 



Tl 



len 



"X 

minus 



(B. 2. pr. 6), that is. 




mmus 
.2 



(B. 3.pr. 18). 
Q. E. D. 



122 BOOK in. PROP. XXXVII. THEOR. 





but 



F from a point outfide of a 
circle tivoftraight lines be 
draivn, the one -■^-■» 
cutting the circle, the 
other — — ^ meeting it, and if 
the reSiangle contained by the whole 
cutting line —"« and its ex- 

ternal fegment ■-..—.. be equal to 
thefquare of the line meeting the circle, 
the latter .m.^m^m,—> is a tangent to 
the circle. 



Draw from the given point 
^— , a tangent to the circle, and draw from the 
centre .«»■», ••».•«.••, and — --- — -^ 
-■^ = X (B.3.pr.36.) 

2 = X (i^yp-). 



and 



Then in ', 





and 



and 



and -,^^^ 



...a»— and 

is common. 



but 



and .*. 



^ = ^ (B. i.pr. 8.); 

^ ^^^ a right angle (B. 3. pr. 18.), 

^r := ^_J| a right angle, 
■^ is a tangent to the circle (B. 3. pr. 16.' 

Q. E. D. 



BOOK IV. 



DEFINITIONS. 

I. 

RECTILINEAR figure is 
faid to be infcribed in another, 
when all the angular points 
of the infcribed figure are on 

the fides of the figure in which it is faid 

to be infcribed. 





II. 

A FIGURE is faid to be defcribed about another figure, when 
all the fides of the circumfcribed figure pafs through the 
angular points of the other figure. 



III. 

A RECTILINEAR figure is faid to be 
infcribed in a circle, when the vertex 
of each angle of the figure is in the 
circumference of the circle. 



IV. 

A RECTILINEAR figure is faid to be cir- 
cumfcribed about a circle, when each of 
its fides is a tangent to the circle. 




124 BOOK IF. DEFINITIONS. 



V. 





A CIRCLE is faid to be tnfcribed in 
a redlilinear figure, when each fide 
of the figure is a tangent to the 
circle. 



VI. 

A CIRCLE is faid to be circum- 
fcribed about a redtihnear figure, 
when the circumference pafles 
through the vertex of each 
angle of the figure. 



y 



is circumfcribed. 




VII. 

A STRAIGHT line is faid to be tnfcribed in 

a circle, when its extremities are in the \ 

circumference. 



The Fourth Book of the Elements is devoted to the folution of J 

problems t chiefly relating to the infcription and circumfcrip- 
tion of regular polygons and circles. 

A regular polygon is one whofe angles and fides are equal. 



BOOK IF. PROP. I. PROP. 



125 




N a given circle 



O 



to place ajlraight line, 
equal to agivenjlraight line ( ), 

not greater than the diameter of the 
circle. 




Draw 



, the diameter of 



and if ..-....^— . ^:z 



', then 



the problem is folved. 



But if 



be not equal to 



(hyp-) ; 



make 



(B. I. pr. 3.) with 
as radius. 



defcribe I ), cutting f |, and 

draw ^ which is the line required. 



For 



(B. I. def. 15. confl.) 



Q. E. D. 



126 



BOOK IF. PROP. II. PROP. 





N a given circle 



O 



to m- 



Jcribe a triangle equiangular 
to a given triangle. 



To any point of the given circle draw 



, a tangent 



(B. 3. pr. 17.) ; and at the point of contadt 
make ^^^ — - ^^ (B. i. pr. 23.) 



and in like manner 
draw 



Ik 



and 



Becaufe 
and 



J^ = ^ (conft.) 
Jg^ = ^^ (B. 3. pr. 32.) 
.*. ^^ = ^U ; alfo 

V^ =: ^r ^°^ ^^ i-3xtit reafon. 

,\^ = ^ (B. i.pr. 32.), 

and therefore the triangle infcribed in the circle is equi- 
angular to the given one. 

Q^E. D. 



BOOK IF. PROP. III. PROB. 



127 




BOUT a given 
circle 



O 



to 



circumfcribe a triangle equi- 
angular to a given triangle. 




Produce any fide 



, of the given triangle both 



ways ; from the centre of the given circle draw 
any radius. 



Make ^ft = 



^ 



and 



(B. I. pr. 23.) 



r=% 



At the extremities of the three radii, draw 



and .-.-...--, tangents to the 
given circle. (B. 3. pr. 17.) 



Zi 



The four angles of >^Wi ^B , taken together, are 

equal to four right angles. (B. i. pr. 32.) 



128 BOOK IF. PROP. III. PROB. 





but ^B ^"d ^^^ ^^^ I'ight angles (confl.) 






two right angles 

but ^H^ ^ La^^^M^ (B' ^' P''- ^3-) 

and = ^^ (conft.) 



and ,*, 

In the fame manner it can be demonftrated that 



<^=^, 



4 = 4 



(B. i.pr. 32.) 

and therefore the triangle circumfcribed about the given 
circle is equiangular to the given triangle. 

Q, E. D. 



i 



BOOK IF. PROP. IV. PROB. 



129 




fcribe a circle. 



Bifeft 



^ and ^V. 



(B. i.pr. 9.) by 
and "— ^^ 



from the point where thefe lines 
meet draw -•■■— , 
and »•••■ refpedlively per- 
pendicular to — BMI^HiB , 







In 



and 



/ 



and 



> 



A 4 



and 



common, ,*, «••••••.•. ^^ .■■■...•.». (B. i. pr. 4and 26.) 

In like manner, it may be fhown alfo 



that 



hence with any one of thefe lines as radius, defcribe 

and it will pafs through the extremities of the 



o 



other two ; and the fides of the given triangle, being per- 
pendicular to the three radii at their extremities, touch the 
circle (B. 3. pr. 16.), which is therefore infcribed in the 



given circle. 



Q. E. D. 



130 



BOOK IV. PROP. V. PROB. 




O defer ibe a circle about a given triangle. 



■" and 



--- (B. I. pr. 10.) 

From the points of bifedlion draw — 
■•■■•■■••• J_ -^■~— ^ and ' 



and 



refpec- 




tively (B. i. pr. 11.), and from their point of 
concourfe draw i^--^^^, «••■-—— and 

and defcribe a circle with any one of them, and 
it will be the circle required. 



In 




(conft.). 




- common, 

^ (conft.), 

(B. I. pr. 4.). 



■■■^•■■aiaKa 



In like manner it may be fhown that 



, , ■■■■««■■■■ ^^ ^■^■^^^^^^" m^^^ "^^^^ \ and 

therefore a circle defcribed from the concourfe of 
thefe three lines with any one of them as a radius 
will circumfcribe the given triangle. 

Q. E. D. 



BOOK IF. PROP. FI. PROB. 131 




O 



N a given circle f j /<? 

infcribe a fquare. 



Draw the two diameters of the 
circle _L to each other, and draw 



o 



is a fquare. 




f 



For, iince ^^^^ and ^^^ are, each of them, in 



a femicirclc, they are right angles (B. 3. pr. 31), 
/. — ^ 11 (B. i.pr. 28): 



and in like manner 





And becaufe mg ^ |^^ (confl.), and 

•••■•■■SM« """ >■■>■■■■■■■ ""• •••»»•■•■••= f B. I . def. I c). 

.*. — = —> — (B. I. pr. 4); 

and fmce the adjacent fides and angles of the parallelo- 

gram ^ X are equal, they are all equal (B. i. pr. 34) ; 



o 



and /, -^ ^ , infcribed in the given circle, is a 
fquare. Q. E. D. 



132 



BOOK IF. PROP. VIL PROP. 





BOUT a given circle 
I i ^^ circumfcribe 



a fquare. 



Draw two diameters of the given 
circle perpendicular to each other, 
and through their extremities draw 



1 9 9 

tangents to the circle ; 



and — 



and 



D 



is a fquare. 




— / I a right angle, (B. 3. pr. 18.) 



alfo 
- II 




(conft.), 
5 in the fame manner it can 



be demonftrated that 
that — ^^ and - 



• •■*••« ■■ !■ 



and alfo 



,», I I is a parallelogram, and 




becaufe 

they are all right angles (B. i. pr. 34) 
it is alfo evident that — --^ , —— ^.^ , 
and -i^— ^ are equal. 



D 



is a fquare. 



Q. E. D. 



BOOK IV. PROP. Fill. PROB. 



133 




O infcribe a circle in a 
given fquare. 



Make 
and 
draw 



and — — II .. 

(B. I. pr. 31.) 





and fince 



is a parallelogram ; 




is equilateral (B. i. pr. 34.) 



In like manner, it can be fhown that 





are equilateral parallelograms ; 



and therefore if a circle be defcribed from the concourfe 
oi thefe lines with any one of them as radius, it will be 
infcribed in the given fquare. (B. 3. pr. 16.) 

CLE. D. 



134 



BOOK IF. PROP. IX. PROB. 





O defcribe a circle about a 
given fquare 



3 



Draw the diagonals ^— — .-. 
and "— ■ interfedting each 

other ; then, 



becaufe 



"^-^Ik 



have 



their fides equal, and the bafe 
■ ■»«*»■• comnion to both, 



or 



r 



^ 



(B. i.pr. 8), 



is bifedled : in like manner it can be (hown 



that 




is bifedted ; 




hence 



^k rr ^^ their halves. 



•. = ; (B. I. pr. 6.) 

and in like manner it can be proved that 



If from the confluence of thefe lines with any one of 

them as radius, a circle be defcribed, it will circumfcrihe 

the given fquare. 

Q. E. D. 



BOOK IV. PROP. X. PROP. 



'35 




O conftruSl an ifofceles 
triangle, in which each of 
the angles at the bafejliail 
he double of the vertical 



angle. 



Take any fliaiwht line ^— 
and divide it fo that 

X = 

(B. 2. pr. I I.) 
With —I"" as radius, defcribe 




o 



in it from the extremity of the radius, 
(B. 4. pr. i) ; draw 



Then 



\ 



and place 



\ is the required triangle. 



For, draw 



and defcribe 



O 



Since 



about ^ I (B. 4. pr. ^.) 

X 



• ■■■■■ X "■ 



•— is a tangent to ( ) (B. 3. pr. 37.) 

.% m = ^ (B. 3. pr. 32), 



1 36 BOOK IV. PROP. X. PROB. 

add ^F to each, 

••• A + < = ^ + ^; 

but ▼ + A or # =z A (B. I. pr. 5) : 
fince -—"m-m ^ ■"»» (B. I. pr. 5.) 

confequently jH[^ ^ ^Xi ^ ^f ^ JH^ 
(B. I. pr. 32.) 

.'. — = (B. i.pr. 6.) 

.•. — ^— =z — — ^— ^ .^_. (conft.) 

.-. -^ = W (B. I.pr. 5.) 

.-. A=^ = A = ^ + 

=: twice y^t 5 and confequently each angle at 
the bafe is double of the vertical angle. 

Q. E. D. 



BOOK IV. PROP. XL PROB. 



137 




N a given circ/e 



o 



to infcribe an equilateral and equi- 
angular pentagon. 

Conftrudl an ifofceles triangle, in 
which each of the angles at the bafe 
fhall be double of the angle at the 
vertex, and infcribe in the given 




▲ 



circle a triangle ^^ equiangular to it ; (B. 4. pr. 2.) 
^ and ^\ (B. i.pr.9.) 



Bifedl 



draw 



and 



Becaufe each of the angles 



A.^.A 



^ 



and \\ are equal, 
the arcs upon which they ftand are equal, (B. 3. pr. 26.) 



and .*. 



and 



....■■». which fubtend thefe arcs are equal (B.3.pr. 29.) 
and ,*, the pentagon is equilateral, it is alfo equiangular, 
as each of its angles ftand upon equal arcs. (B. 3. pr. 27). 

Q^E. D. 



138 



BOOK IV. PROP. XII. PROP. 





O defcribe an equilateral 
and equiangular penta- 
gon about a given circle 



O- 



Draw five tangents through the 
vertices of the angles of any regular 
pentagon infcribed in the given 



circle 



o 



(B. 3. pr. 17). 

Thefe five tangents will form the required pentagon. 



Draw 






' 1:™ 



In 




and 



■ ^■•■■■■■B 



(B. i.pr.47), 

and ■ common ; 



.-.7 = 



<- 



twice 



and ▼ = .4. (B. i.pr. 8.) 

, and ^^1 =r twice ^ 
In the fame manner it can be demonilrated that 

^^/ =: twice ^^ , and ^r ^ t^vice ^; 

but ^ ^ " B. 3.pr. 27), 



BOOK IV. PROP. XII. PROB. 139 

their halves = j^ , alfo £ I ^ I \ 9 and 

-■■■-■■■» common ; 

and «-i-iaMaiii. ^ ...HMMiB, 

,•, •■^-■» .— ^K ^ twice — ^— 5 

In the fame manner it can be demonftrated 

that 1^^^---— ^ twice ■-■^-•, 

but — — = — — • 

In the fame manner it can be demonftrated that the 
other fides are equal, and therefore the pentagon is equi- 
lateral, it is alfo equiangular, for 

^^ ^ twice 1^^ and \^^ = twice j^^ , 



and therefore 

• mKkl — uflB 9 1" the fame manner it can be 
demon ftrated that the other angles of the defcribed 
pentagon are equal. 

QE. D 



140 



BOOK IF. PROP. XIII. PROP. 




Draw 



Becaufe 
and 




O infcribe a circle in a 
given equiangular and 
equilateral pentagon. 




^^^ «■/ ^^ ^ given equiangular 

and equilateral pentagon ; it is re- 
quired to infcribe a circle in it. 



Make 



^=^,andi|^=^ 
(B. i.pr. 9.) 



= - ,r=A, 

common to the two triangles 



&c. 




/ 



and >A ...,.lk ; 

.. and ^r ^ J|^ (B. i. pr. 4.) 



And becaufe ^^ ^ 



,*, r= twice 



4 



twice 



is bifedled by 



In like manner it may be demonftrated that 



^ 



IS 



«••• "j and that the remaining angle of 



bifedled by 

the polygon is bifedled in a fimilar manner 



BOOK IV. PROP. XIII. PROB. 141 

Draw ^i— -i^ , -....-.. , 6cc. perpendicular to the 
fides of the pentagon. 



Then in the two triangles ^^ and 




A 



we have ^^ z= ^^^,(conft.), ^^^^i^ common, 
and ^V :^ JIh = a right angle ; 



, (B. I. pr. 26.) 



In the fame way it may be fhown that the five perpen- 
diculars on the fides of the pentagon are equal to one 
another. 



o 



Defcribe X^ ^ with any one of the perpendicu- 
lars as radius, and it will be the infcribed circle required. 
For if it does not touch the fides of the pentagon, but cut 
them, then a line drawn from the extremity at right angles 
to the diameter of a circle will fall within the circle, which 
has been fhown to be abfurd. (B. 3. pr. 16.) 

f^E. D. 



14* 



BOOK IV, PROP. Xn\ PROB. 




pO dcfirihc j r.-TiV chcn: s 



grom egh:.s: 



'sJ ^nd csiii 



oik and ^^ 



Bilect ^JHk and 
bT ••••••»»»«» and ..-•...... , and 

^om the point of fedion, draw 

._^B , >••»»• , and ^^^^ . 



(B. i.pr.6): 




I" like manner it mar be proved that 
■ ^ ^iB^^M ^ ^^— — , and 
therefore -••••••• ^ — ^— ^ ••m>»...w. 



Therefore if a circle be defcribed from the point where 
thefe five lines meet, with any one of tfaem 

as a radius, it will circumicribe 

the given pcntagoo. 

Q E- D. 




BOOK W. PROP. XV. PROB. 

O infcribe an equilateral and equian- 
gular hexagon in a gircen circle 



H3 



O- 



From any point in the circumference of 
the given circle defcribe ^ J palling 



o 



through its centre, and draw the diameters 



and 



draw 



...._»■_ J .-..-..-^ .........J &c. and the 

required hexagon is inicribed in the given 
circle. 




Since 



of the circles. 



palles through the centres 




and 




are equilateral 



4 = ^ 



triangles, hence ^^ ^ ^^ ^ one-third of two right 




angles; i^B. i. pr. 32) but 

(B. i.pr. 13); 



= m 



^ one-third of 



£Di 



(B. I. pr. 32% and the angles vertically oppoiite re :::ei"e 
are all equal to one another (B. i. pr. i ;\ and iland on 
equal arches (B. 3. pr. 26), which are fubtended by equal 
chords (B. 3. pr. 29) ; and fince each of the angles of the 
hexagon is double of the angle of an equilateral triangle, 
it is alio equiangular. O ^ F) 



'44 



BOOK IV. PROP. XVI. PROP. 




O infcribe an equilateral and 
equiangular quindecagon in 
a given circle. 



and 



be 



the fides of an equilateral pentagon 
infcribed in the given circle, and 
««»-— the fide of an inscribed equi- 
lateral triangle. 



The arc fubtended by 
. and __ 



_6_ 
I 4 



of the whole 
circumference. 



The arc fubtended by 1 



_5_ 
1 i 



Their difference =: tV 



,'. the arc fubtended by 
the whole circumference. 



of the whole 
circumference. 



zz. tV difference of 



Hence if firaight lines equal to ..-«.—■« be placed in the 
circle (B. 4. pr. i), an equilateral and equiangular quin- 
decagon will be thus infcribed in the circle. 

Q. E. D. 



BOOK V. 



DEFINITIONS. 



I. 




LESS magnitude is faid to be an aliquot part or 
fubmultiple of a greater magnitude, when the 
lefs meafures the greater ; that is, when the 



'^ lefs is contained a certain number of times ex- 



adlly in the greater. 



II. 



A GREATER magnitude is faid to be a multiple of a lefs, 
when the greater is meafured by the lefs ; that is, when 
the greater contains the lefs a certain number of times 
exadlly. 

III. 

Ratio is the relation which one quantity bears to another 
of the fame kind, with refpedl to magnitude. 



IV. 

Magnitudes are faid to have a ratio to one another, when 
they are of the fame kind ; and the one which is not the 
greater can be multiplied fo as to exceed the other. 

TAe of her definitions will be given throughout the book 
where their aid is Jirjl required. 
u 



146 



AXIOMS. 




QUIMULTIPLES or equifubmultiples of the 
fame, or of equal magnitudes, are equal. 

If A = B, then 
twice A ^ twice B, that is, 
2 A = 2 B; 
3Az=3B; 
4 A = 4B; 
&c. &c. 
and i of A = i of B ; 
i of A = i of B ; 
&c. &c. 

II. 

A MULTIPLE of a greater magnitude is greater than the fame 
multiple of a lefs. 

Let A C B, then 

2 AC 2 B; 

3 ACZ3B; 

4 AIZ4B; 

&c. &c. 

III. 

That magnitude, of which a multiple is greater than the 
fame multiple of another, is greater than the other. 

Let 2 A C 2 B, then 

ACB; 
or, let 3 A C 3 B, then 

ACB; 
or, let w A CZ m B, then 

ACB. 



\ 



BOOK V. PROP. I. THEOR. 



H7 




F any number of magnitudes be equimultiples of as 

many others, each of each : what multiple soever 

any one of the firjl is of its part, the fame multiple 

jhall of the fir ft magnitudes taken together be of all 

the others taken together. 

LetQQQQQ be the fame multiple of Q, 
that Pip^^^ isof ^. 
that OOOQO ^s of Q. 



Then is evident that 

• QQQQQ 



OOOOQ 



fQ 



is the fame multiple of < 



Q 



which that QQQQQ is of Q ; 
becaufe there are as many magnitudes 



QQQQQ 1 



m <! 






.QOOQOJ 



Q 
Q 



as there are in QQQQQ := Q. 

The fame demonftration holds in any number of mag- 
nitudes, which has here been applied to three. 



,*, If any number of magnitudes, &c. 




148 BOOK r. PROP. 11. THEOR. 



|F the fir ft magnitude be the fame multiple of the 
fecondthat the third is of the fourth, and the fifth 
the fame multiple of the fecond that the fixth is oj 
the fourth, then fiall the firfi, together with the 

fifth, be the fame multiple of the fecond that the third, together 

•with the fixth, is of the fourth. 

Let ^01 9, the firil, be the fame muhiple of ^, 
the fecond, that OO 0> ^'^^ third, is of <^, the fourth ; 
and let 9 0, the fifth, be the fame multiple of * , 
the fecond, that OOOOj ^^^ fixth, is of <2>, the 
fourth. 

Then it is evident, that \ .^ ,^. ,^. .^ ' , the firfl and 
fifth together, is the fame multiple of , the fecond, 
that \ !r!r^^ k the third and fixth together, is of 

looooj 

the fame multiple of <2>) the fourth ; becaufe there are as 

f #•• 1 
many magnitudes i" ] ^^^^ ^ 3P ^s there are 



• f 000 \ _ ^ 



,*, If the firfl: magnitude, &c. 



BOOK V. PROP. III. THEOR. 



149 




F thefirjl of four magnitudes be the fame multiple 
of the fecond that the third is of the fourth, and 
if any equimultiples whatever of the firjl and third 
be taken, thofe Jliall be equimultiples ; one of the 



fecond, and the other of the fourth. 



The First. 



The Second. 



Let \ " |- be the iame multiple of 

I ! 

The Third. The Fourth. 

which j T T [ is of A ; 



take <;^ S S S S > the fame multiple of < 



which \ J 3 ? A 

[♦♦♦♦ 



> is of \ 



♦♦ 



that ^ 



Then it is evident, 

Tlie Second. 

is the fame multiple of | 



150 



BOOK V. PROP. III. THEOR. 



which i 



♦♦♦♦ 



♦ ♦♦♦ 

♦ ♦♦♦ 



The Fourth. 

- is of ^ ; 



J 



becaufe < 



> contains 



> contains 



♦♦♦♦' 

♦♦♦♦ 

♦♦♦♦ 



as many times as 



;- contains ■! T^ T^ !> contains ^ 



♦♦ 



The fame realbning is applicable in all cafes. 



.*. If the firft four, &c. 



BOOK V. DEFINITION V. 



'51 



DEFINITION V. 

Four magnitudes, ^^ 01 ^ ^ > ^j are faid to be propor- 
tionals when every equimultiple of the firft and third be 
taken, and every equimultiple of the fecond and fourth, as. 



of the firfl 



&c. 

of the fecond 



of the third ^ ^ 



♦ ♦♦♦ 
♦ ♦♦♦♦ 

♦♦♦♦♦♦ 

&c. 
of the fourtli 



&c. &c. 

Then taking every pair of equimultiples of the firft and 
third, and every pair of equimultiples of the fecond and 
fourth, 

' — =or^«« 

= or 31 

or ^ 

= or 33 

= or — 1 



r 



\^<mm cz, = 



L 





^♦4 C. = °r 




♦ ♦ C. = or 


then will - 


♦ ♦ C, = or 




tt C. = or 




^♦^ C. = or 



152 



BOOK F. DEFINITION V. 



That is, if twice the firfl be greater, equal, or lefs than 
twice the fecond, twice the third will be greater, equal, or 
lefs than twice the fourth ; or, if twice the firfl be greater, 
equal, or lefs than three times the fecond, twice the third 
will be greater, equal, or lefs than three times the fourth, 
and so on, as above expreffed. 



If 



or 
or 
or 
or 
or 



then 
will 



&c. 

'♦♦♦ 
♦♦♦ 
♦♦♦ 

♦ ♦♦ 

♦ ♦♦ 

&c. 



c = 



6cc. 



or Z] 

or 31 

or ^ 

or Z] 

or ^ 



&c. 



In other terms, if three times the firft be greater, equal, 
or lefs than twice the fecond, three times the third will be 
greater, equal, or lefs than twice the fourth ; or, if three 
times the firft be greater, equal, or lefs than three times the 
fecond, then will three times the third be greater, equal, or 
lefs than three times the fourth ; or if three times the firft 
be greater, equal, or lefs than four times the fecond, then 
will three times the third be greater, equal, or lefs than four 
times the fourth, and so on. Again, 



BOOK V. DEFINITION V. 



^Sl 



If < 



Sec. 



then 
will 



♦ ♦♦♦ 

♦ ♦♦♦ 



^ or ^ 

= or ^ 

= or Zl 

=: or ID 

= or ID 

= or Zl 

= or Zl 

= or Z] 

= or Zl 

= or Zl 



&c. 



&c. 



&;c. 



And so on, with any other equimultiples of the four 
magnitudes, taken in the fame manner. 

Euclid exprefles this definition as follows : — 

The firft of four magnitudes is faid to have the fame 
ratio to the fecond, which the third has to the fourth, 
when any equimultiples whatfoever of the firft and third 
being taken, and any equimultiples whatfoever of the 
fecond and fourth ; if the multiple of the firft be lefs than 
that of the fecond, the multiple of the third is alfo lefs than 
that of the fourth ; or, it the multiple of the firft be equal 
to that of the fecond, the multiple of the third is alfo equal 
to that of the fourth ; or, ii the multiple of the firft be 
greater than that of the fecond, the multiple of the third 
is alfo greater than that of the fourth. 

In future we ftiall exprefs this definition generally, thus : 

If M # C = or Zl ''^ , 
when M ^ C = or 313 ;;; ^ 



154 BOOK V. DEFINITION F. 

Then we infer that , the firft, lias the lame ratio 
to ^ , the fecond, which ^ , the third, has to ^ the 
fourth : exprelTed in the fucceeding demonftrations thus : 

# :ii :: 4 : V; 

or thus, : It = ^ • V 7 

or thus, — ^ =■ : and is read, 

" as is to , so is ^ to ^. 

And if # : " : : ^ : ip we fhall infer if 

M C5 ^= or ^ w , , , then will 

M ^ C = or 13 w ^. 

That is, if the firfl; be to the fecond, as the third is to the 
fourth ; then if M times the firft be greater than, equal to, 
or lefs than m times the fecond, then (hall M times the 
third be greater than, equal to, or lefs than m times the 
fourth, in which M and m are not to be confidered parti- 
cular multiples, but every pair of multiples whatever; 
nor are fuch marks as 0, ^, , &c. to be confidered 
any more than reprefentatives of geometrical magnitudes. 

The ftudent fhould thoroughly underftand this definition 
before proceeding further. 



BOOK V. PROP. IF. THEOR. 



^SS 



F the firjl of four magnitudes have the fame ratio to 
the fecond, which the third has to the fourth, then 
any equimultiples whatever of the frji and third 
shall have the fame ratio to any equimultiples of 
the fecond and fourth ; viz., the equimultiple of the firji Jliall 
have the fame ratio to that of the fecond, which the equi- 
multiple of the third has to that of the fourth. 



m 



Let : ■ ::^ :^, then3 :2|::3^:2^, 

every equimultiple of 3 and 3 ^ are equimultiples of 
^ and ^ , and every equimultiple of 2 ^ and 2 ^ , are 
equimultiples of | and ^ (B. 5, pr. 3.) 

That is, M times 3 '^ and M times 3 ^ are equimulti- 
ples of and ^ , and ;;z times 2 | and w 2 1^ are equi- 
multiples of 2 H and 2 ^ ; but • H • • ^ • V 

(hyp); .*. if M 3 C =, or ;^ »? 2 |||, then 

M 3 ^ C r=, or Z] « 2 ip (def. 5.) 

and therefore 3 ^:2||::3^:2^ (def. 5.) 

The fame reafoning holds good if any other equimul- 
tiple of the firft and third be taken, any other equimultiple 
of the fecond and fourth. 

,*. If the firfl: four magnitudes, &c. 



156 



BOOK V. PROP. V. THEOR. 




F one magnitude be the fame multiple of another, 
which a magnitude taken from thefirfl is of a mag- 
nitude taken from the other, the remainder Jhall be 
the fame multiple of the remainder, that the whole 



is of the whole. 



Q 

LetQQ 
O 



= M'^ 



and 



= M-,, 



o 

C^<^ minus = M' minus M' «> 

D 

.-. <> =M'(Jminus.), 



and /. ^ = M' A. 



/, If one magnitude, Sec. 



BOOK V. PROP. VI. THEOR. 157 




F two magnitudes be equimultiples of two others, 
and if equimultiples of thefe be taken from the fir Ji 
two, the remainders are either equal to thefe others, 
or equimultiples of them. 



Q 

Let :yQ = M' ■ ; and QQ = M' a ; 

o 

Q 

then 00 minus ni m ::^ 

o 

M' ■ minus /w' « = (M' minus m') b, 

and 00 minus m' k := M' a minus /«' 4 := 
(M' minus m') k . 

Hence, (M' minus ;«') m and (M' minus tn') k are equi- 
multiples of K and k , and equal to * and a 9 
when M' minus m' ':^i i. 

.'. If two magnitudes be equimultiples, &c. 



158 



BOOK F. PROP. A. THEOR. 




F the firjl of t/ie four magnitudes has the fame ratio 
to the fecond which the third has to the fourth, 
then if the firjl be greater than the fecond, the 
third is alfo greater than the fourth ; and f equal, 
equal ; if lefs, lefs. 

Let ^ : H : r ip : ^ ; therefore, by the fifth defini- 
tion, if %% d ■■, then will ^^ C #4 ; 
but if # CZ ■, then ## [Z ■■ 
and ^fp [= 
and .*. ^ C ;► . 

Similarly, if ^ ^, or ^ J, then will ^ z^, 
or ^ . 

.'. If the firfl of four, &c. 



DEFINITION XIV. 

Geometricians make ufe of the technical term " Inver- 
tendo," by inverfion, when there are four proportionals, 
and it is inferred, that the fecond is to the firfl as the fourth 
to the third. 

Let A : B : : C : O, then, by " invertendo" it is inferred 
B : A :: I) : C. 



BOOK V. PROP. B. THEOR. 



'50 




F Jour magnitudes are proportionals , they are pro- 
portionals alfo when taken inverfely. 



Let ^ : O : : ■ : ^ , 

then, inverfely, O : ^ 1 1 : ■ . 

If M ^ n « O? then M ■ I] w ^ 
by the fifth defimtion. 

Let M ^ ID /w Q, that is, w Q CZ M ^ , 
.*. M B lU w , or, /« CZ M ■ ; 
.*. if w O CZ M ^, then will w C M B 

In the fame manner it may be (liown, 

that if ;« Q := or Z] M ^ , 

then will m :=, or 13 M B ; 

and therefore, by the fifth definition, we infer 

that O : ^ : '^ : H . 
.', If four magnitudes, &c. 



i6o 



BOOK V. PROP. C. THEOR. 




F the fiyji be the fame multiple of the fecond, or the 
fame part of it, that the third is of the fourth ; 

the frjl is to the fecond, as the third is to the 

fourth. 



Let ^ ^ t the firfl:,be the fame multiple of ^, the fecond, 
that 7 J, the third, is of ■, the fourth. 



♦ ♦.4 

♦ ♦ 



,m 0, M ? ?,>« A 



Then _ _ 

takeMj J 

■ ■ 
becaufe^S is the fame multiple of ^ 

that J J is of 4 (according to the hypothcfis) ; 

and ^M^ is taken the fame multiple ofSS 
that M T T is of ? T , 

,*, (according to the third propolition), 
M ^ _ is the fame multiple of ^ 

that M T T is of 4. 



BOOK F. PROP. C. THEOR. i6i 



Therefore, if M ^ ^ be of ^ a greater multiple than 

;// ^ is, then M J i is a greater multiple of ^ than 

w A is ; that is, if M S S ^^ greater than m 0, then 

M J J will be greater than m ^ ; in the fame manner 

it can be fliewn, if M ^ ^ be equal m ^^ then 

M J J will be equal m A. 



And, generally, if M ^ ^ C =z or ^ //; 
then M will be C ^ or ^ ;« ^* 

,', by the fifth definition, 

■ ■_..♦♦.▲ 

■ ■•••♦♦•■• 

■ ■ 
Next, let be the fame part of J S 

that itk is of T T- 
In this cafe alfo : J J :: (ffc : TT. 



For, becaufe 

■■"*■"-""' WW 



is the fame part of ^ ^ that ■ is of ^ ^ , 



1 62 BOOK F. PROP. C. THEOR. 

therefore S S is the fame muhiple of 

that ^ J is of ^ . 

Therefore, by the preceding cafe, 

■ ■ . A .. . ^ . 

and • ^ • ■■ •• A • ^^ 

by propofition B. 
/. If the firfl be the fame multiple, &c. 



BOOK V. PROP. D. THEOR. 



163 




\^ the fir Jl be to the fecond as the third to the fourth, 
and if the firfi be a multiple, or a part of the 
fecond; the third is the fame multiple, or the fame 
part of the fourth. 



and firft, let 



be a multiple H; 



J J fhall be the fame multiple of ■. 

First. Second. Tliird. Fourth. 

QQ 00 

Take ^^ =r ^ 
QQ 



Whatever multiple 
take ^^ 
then, becaufe 



is of I 
the fame multiple of 1 
• ....♦♦ 



and of the fecond and fourth, we have taken equimultiples, 

^nd )f Y , therefore (B. c. pr. 4), 
00 



i64 BOOK F. PROP. D. THEOR. 

'OCl"^^ . ^^, but (conft.), 

and y\y\ is the fame multiple of ■ 
that ^ is of U . 

Next, let B : ^ ^ • ■ V • T J » 

and alfo H a part of -^^ ; 

then ip fhall be the fame part of J J , 

Inverfely (B. 5.), ^ ' ■ *= ^ J ' V' 
but I is a part of ^^ ; 
that is, ^ ^ is a multiple of | ; 

, by the preceding cafe, X X is the fame multiple of ^ 
that is, ^ is the fame part of X X 
that H is of 

,% If the firft be to the fecond, &c. 



BOOK V. PROP. VII. THEOR 



165 




QUAL magnitudes have the fame ratio to the fame 
tnagnitiide, and the fame has the fame ratio to equal 
magnitudes. 



Let ^ = ^ and any other magnitude ; 

then : = ♦ •* and : # = : : ^ , 

Becaufe ^ ^ ^, 
.-. M # = M ^ ; 

/, if M C := or [3 /;/ , then 

M ^ C = or ;^ ;« , 
and .*. % : c = ♦ : ■ (B. 5. def. 5). 

From the foregoing reafoning it is evident that, 
i£ m C > = Of ZD M , then 

wHC^orl^ M^ 
/.■:#=■:♦ (B. 5. def. 5). 

,*. Equal magnitudes, &c. 



i66 BOOK F. DEFINITION VII. 



DEFINITION VII. 

WiiKN of the equimultiples of four magnitudes (taken as in 
tile fifth definition), the multiple of the firft is greater than 
tli:it of the fecond, but the multiple of the third is not 
greater than the multiple of the fourth ; then the firft is 
laid to have to the fecond a greater ratio than the third 
magnitude has to the fourth : and, on the contrary, the 
third is faid to have to the fourth a lefs ratio than the firft 
has to the fecond. 

If, among the equimultiples of four magnitudes, com- 
pared as in the fifth definition, we fhould find 

• #••# [=■■■■, but 
44^44 = '"' ^ W IP V. or if we mould 
rnul .iny particular multiple M oi the firft and third, and 
a particular multiple m' of the fecond and fourth, fuch, 
that M times the firft is C w' times the fecond, but M' 
times the third is not CZ w times the fourth, i.e. ^ or 
~~1 "; times the fourth ; then the firll is faid to have to 
the tiwnd a strcater ratio than the third has to the fourth; 
v>r the thial has to the fourth, under fuch circumftances, a 
lets ratio than the hrtl has to the feccaid : although feveral 
other equimultiples may tend to ibow that the four mag- 
nitudes arc piv>portionAls. 

This det\i\itivM\ will in tuiure be exprdSbd ti^is : — 

ItM fP C rr Q. but M ■ = - Z3 'T ♦ , 

then P : ~ IZ ■ : ♦ . 

In the aK>\^ cv' ;- ' ;\ -tSoia* M «ad af aie to be 
wnikkitHi jvftrtkn;. cs, at* fike dK iMilli|ilr' M 



BOOK F. DEFINITION VII. 



167 



and m introduced in the fifth definition, which are in that 
definition confidered to be every pair of multiples that can 
be taken. It muft alfo be here obferved, that ip , U, H , 
and the like fymbols are to be confidered merely the repre- 
fentatives of geometrical magnitudes. 

In a partial arithmetical way, this may be fet forth as 
follows : 

Let us take the four numbers, 8 , 7, j c , and 



FirJi. 


Second. 


Third. 


Fourth. 


8 


7 


10 


Q 


lO 


14 


2C» 




24 


21 


30 


^7 


32 


28 


40 


36 


40 


35 


50 


45 


48 


42 


60 


54 


56 


49 


70 


63 


64 


56 


80 


72 


72 


63 


90 


8t 


80 


70 


lOD 


".'- 


88 


77 


no 


vy 


96 


84 


120 


108 


104 


9' 


'3° 


117 


T12 


98 


140 


126 


&c. 


&c. 


&c 


&c. 



Among the above multiples we find 16 C 14 and 20 
r~ that is, twice the firft is greater than twice the 

fecond, and twice the third is greater than twice the fourth ; 
and 16^21 and 20 "^ that is, twice the firil is lefs 

than three times the fecond, and twice the third is lefs than 
three times the fourth ; and among the fame multiples we 
can find -: C 56 and V - C that is, 9 times the firft 

is greater than 8 times the fecond, and 9 times the third is 
greater than 8 times the fourth. Many other equimul- 



1 68 BOOK V. DEFINITION VII. 

tiples might be selected, which would tend to Ihow that 
the numbers %,y, \o, were proportionals, but they are 
not, for we can find a multiple of the firlt ^ a multiple of 
the fecond, but the fame multiple of the third that has been 
taken of the firft not C the fame multiple of the fourth 
which has been taken of the fecond ; for inftance, 9 times 
the hrll: is C i o times the fecond, but 9 times the third is 
not C ^° times the fourth, that is, -: C 70, but 90 
not CZ or 8 times the firfl we find C 9 times the 

fecond, but 8 times the third is not greater than 9 times 
the fourth, that is, O-i-C 63, but Sc is not C When 

any fuch multiples as thefe can be found, the hrft (3~)is 
faid to have to the fecond (7) a greater ratio than the third 
(10) has to the fourth and on the contrary the third 

(10) is faid to have to the fourth a lefs ratio than the 
firfl (3) has to the fecond (7). 



BOOK r. PROP. Fill. THEOR. 



109 




F unequal magnitudes the greater has a greater 
ratio to the fame than the lefs has : and the fame 
magnitude has a greater ? atio to the lefs than it 
has to the greater. 



Let m and be two unequal magnitudes, 
and ^ any other. 

k 
We fliall firft prove that H which is the greater of the 

two unequal magnitudes, has a greater ratio to than , 

the lefs, has to ^ ^ 

that is, ■ : [Z , : # ; 

take M' l^/^' #, M' ■, and tn % ; 

fuch, that M' ▲ and M' H fhall be each C # ; 

alfo take ;;/ ^ the leaft multiple of ^ , 



which will make m 



M' 



= M' 



.*. M' is not CZ f" 



butM' 



IS 



?n 



for. 



as m' A is the firft multiple which firft becomes C M'^, 

than (w minus I ) ^ or;;/ minus ^ isnotCM' JU, 

and ^ is not CI M' a, 

/. ;;;' minus + # "^"^ be Zl M' Jj + M' A ; 

A 

that is, ;;;' % mull be i;;^ M' ■ ; 



.-. M' 



IS 



tn 



', but it has been fhown above that 



170 BOOK F. PROP. Fill. THEOR. 

M' m is note »?' # , therefore, by the feventh definition, 
m has to a greater ratio than 1:0. 

Next we fhall prove that % has a greater ratio to ^ , the 
lefs, than it has to j^ , the greater ; 

o''# :■ [= • :■• 

A 

Take /;/ 0, M' ■■, m' #, and M' ||, 

the fame as in the firll: cafe, fuch, that 

M' A and M' jp will be each C , and m % the leail 

multiple of ^ , which firft becomes greater 

than M' H = M' || . 

.'. m ^ minus ^ is notC M' ^, 

and ^ is not CI M' A ; confequently 

m % minus # -}- # is ZH M' g -f M' a ; 

▲ 

,*, m' ^ is ID M' ■, and ,'. by the feventh definition, 

A 
^ has to a| ^ greater ratio than ^ has to ■ . 

,'. Of unequal magnitudes, &c. 

The contrivance employed in this propofition for finding 
among the multiples taken, as in the fifth definition, a mul- 
tiple of the firfl greater than the multiple of the fecond, but 
the fame multiple of the third which has been taken of the 
firft, not greater than the fame multiple of the fourth which 
has been taken of the fecond, may be illuftrated numerically 
as follows : — 

The number 9 has a greater ratio to 7 than has to 7 : 
that is, 9 : 7 C : 7 ; or, 8 + i : 7 C = 7- 



BOOKF. PROP. Fill. THEOR, 171 

The multiple of i , which firft becomes greater than 7, 
is 8 times, therefore we may multiply the firft and third 
by 8, 9, 10, or any other greater number; in this cafe, let 
us multiply the firft and third by 8, and we have '^-^-f- 8 
and : again, the firft multiple of ^ which becomes 
greater than 64 is 10 times; then, by multiplying the 
fecond and fourth by 10, we ftiall have 70 and 70 ; then, 
arranging thefe multiples, we have — 

8 times lo times 8 times lo times 

the first. the second. the third. the fourtli. 

6^+ 8 -0 -o 

Confequently 04 -j- 8, or 72, is greater than -o, but -t^ 
is not greater than 70, .•. by the feventh definition, 9 has a 
greater ratio to 7 than has to 7 . 

The above is merely illuftrative of the foregoing demon- 
ftration, for this property could be fhown of thefe or other 
numbers very readily in the following manner ; becaufe, if 
an antecedent contains its confequent a greater number of 
times than another antecedent contains its confequent, or 
when a fraction is formed of an antecedent for the nu- 
merator, and its confequent for the denominator be greater 
than another fraction which is formed of another antece- 
dent for the numerator and its confequent for the denomi- 
nator, the ratio of the firft antecedent to its confequent is 
greater than the ratio of the laft antecedent to its confe- 
quent. 

Thus, the number 9 has a greater ratio to 7, than 8 has 
to 7, for ^ is greater than -. 

Again, 17 : 19 is a greater ratio than 13 : 15, becaufe 
17 17 X 15 _ 255 J 13 13 X 19 247 , 

evident that ^ is greater than |g, .-. J-^ is greater than 



1/2 BOOK F. PROP. VIIT. THEOR. 



— , and, according to wliat has been above fliown, 17 has 
to 19 a greater ratio than 13 has to 15. 



So that the general terms upon which a greater, equal, 
or lefs ratio exifVs are as follows : — 

A C . . 

If g be greater than ^, A is faid to have to B a greater 

A C 

ratio than C has to D ; if — be equal to rr, then A has to 
B the fame ratio which C has to D ; and if -^ be lefs than 
^, A is faid to have to B a lefs ratio than C has to D. 

The ftudent fhould underftand all up to this propofition 
perfectly before proceeding further, in order fully to com- 
prehend the following propofitions of this book. We there- 
fore ftrongly recommend the learner to commence again, 
and read up to this flowly, and carefully reafon at each ftep, 
as he proceeds, particularly guarding againlT; the mifchiev- 
ous fyftem of depending wholly on the memory. By fol- 
lowing thefe inftrudions, he will find that the parts which 
ufually prefent confiderable difficulties will prefent no diffi- 
culties whatever, in profecuting the ftudy of this important 
book. 



BOOK V. PROP. IX. THEOR. 



^71, 




AGNITUDES which have the fame ratio to the 
fame magnitude are equal to one another ; and 

thofe to which the fame magnitude has t/ie fame 

rat to are equal to one another. 

Let ^ : ^ : : : p, then ^ = . 
For, if not, let ▲ CI ? then will 

4 : «^ C # : (B. 5- pr. 8), 

which is abfurd according to the hypothefis. 
.*. ^ is not C . 

In the fame manner it may be fhown, that 
A is not ^ ▲, 

/. 4 =#. 

Again, let H : ^ : : '^' : ^ , then will ^ = . 

For (invert.) ^ : || : : f| : H, 
therefore, by the firft cafe, A ^ A . 

,*. Magnitudes which have the fame ratio, &c. 

This may be {hown otherwife, as follows : — 
Let ^ : B ^ A : C, then B = C, for, as the fradlion 
— = the fradlion -, and the numerator of one equal to the 
numerator of the other, therefore the denominator of thefe 
fradlions are equal, that is K zz C. 

Again, if B : ,\ = C : A , B = C. For, as - = "^, 
B muft = (,. 



174 



BOOK V. PROP. X. THEOR. 




HAT magnitude which has a greater ratio than 
another has unto the fame magnitude, is the greater 
of the two : and that magnitude to which the fame 
has a greater ratio than it has unto another mag- 
nitude, is the lefs of the two. 

Let ^ : C # : ■> then ^ d # . 

For if not, let |p =: or ^ ; 

then, ^ : si = # : B (^- 5- P^- l) or 

^ : H 13 ^ : ■ (B. 5. pr. 8) and (invert.), 

which is abfurd according to the hypothefis. 

,*, ■ is not =: or ^ ^ , and 
.'. S muft he r~ ^. 

Again, let «: C V : fP, 
then, ID ^> 

For if not, mufl: be C or ^ 1^ , 

then flj: Z] p: ^ (B- 5. pr. 8) and (invert.); 

or fl: =: H* V (B. 5. pr. 7), which is abfurd (hyp.); 

/. is not CZ or = ^ , 

and .'. mufl be ^ ^ . 

.*. That magnitude which has, &c. 



BOOK V. PROP. XL THEOR. 



^75 




ATIOS t/iat are the fame to the fame ratio, are the 
fame to each other. 



Let ^ : ■ = : IP' and : P = ▲ : •, 
then will ^ : H = ▲ : •. 

For if M ^ C =, or 13 m H, 

then M IZ> ^. or 3] w ^, 

and if M CZ, ^, or ^ ;/; t' , 

then M A C =, or Z3 m •, (B. 5. def. 5) ; 

, if M ^ C, ^, or ^ w d 9 M A [Z, =, or Zl ^« •> 
and .*. (B. 5. def. 5) ^ I H ^ A : •. 

.*, Ratios that are the fame, &c. 




176 BOOK r. PROP. XII. THEOR. 



F afijf number of tnagnitiides be proportionals, as 
one of the antecedents is to its confeqiient, fo f}:>all 
all the antecedents taken together be to all the 
confequents. 

then will | : # = 
■ +0+ +« + ^:# + 0+ +' + •• 

For if M U IZ w 0, then M Q [Z w <>, 

and M \^tn M • CZ ^« t, 

alfoM ▲ IZ'« •• (B. 5. def. 5.) 

Therefore, if M JH C w 0, then will 

MJ+MQ + M -I-M. + Ma, 

or M (H + O + + • + ^) be greater 

tlian ;;/ ^ •\- tn ^ -\- m •\' ^'^ ▼ "h ^^^ •> 

or;^(# +0+ , + ^ + *)- 

In the fame way it may be fhown, if M times one of the 
antecedents be equal to or lefs than m times one of the con- 
fequents, M times all the antecedents taken together, will 
be equal to or lefs than ni times all the confequents taken 
together. Therefore, by the fifth definition, as one of the 
antecedents is to its confequent, fo are all the antecedents 
taken together to all the confequents taken together. 

.*, If any number of magnitudes, &c. 



BOOK V. PROP. XIII. THEOR. 



'77 




F the firjl has to the fecond the fame ratio which 

the third has to the fourth, but the third to the 

fourth a greater ratio than the fifth has to the 

fixth ; the firfi fhall afo have to the fecond a greater 

ratio than the fifth to the fixth. 

Let fP : O = ■ : # , but ■ : A ci O : #, 

then fP : D IZ O : •• 

For, becaufe | : d O • 0> there are fome mul- 

tiples (M' and ;«') of | and ^^ and of ^ and ^, 



fuch that M' 



m 



but M' <^ not C m 0, by the feventh definition. 

Let thefe multiples be taken, and take the fame multiples 
of ■ and ([n. 
/. (B. 5. def. 5.) if M' ^ C, =, or Zl /«' Q ; 
then will M' | C =, or ^ m , 
but M' ■ C »^' ^ (conftrudlion) ; ■ 

.*. M' ^ C ni Q , 

but M' <3 is not C ni (conflrudlion) ; 
and therefore by the feventh definition, 

W :0 CIO 



^^v * 



.*. If the firll; has to the fecond, &c. 



A A 



178 



BOOK V. PROP. XIV. THEOR. 




F the fir Jl has the fame ratio to the fecondivhich the 
third has to the fourth ; then, ifthefirjl be greater 
than the third, thefecondjhall be greater than the 
fourth; and if equal, equal; andiflefs, lefs. 



Let ^ : Q : : B : ^ , and firfl fuppofe 
IP [Z » , then will Q C ^ • 

For^rQCI : IJ (B.5.pr. 8), andbythe 
hypothefis, ^ : O = ^Ji : ^ ; 
/. ■ : ♦ CZ :D(B. s-pr-'is). 

/. ♦ Z3 D (B- 5- pr- io-)» or Q C ♦• 
Secondly, let ^ = |P , then will ^ ^ ^ . 

For ^ : O = : D (B. 5. pr. 7), 
and ^ : Q = : ^ (hyp.) ; 

.*. ■ : D= -V : ♦ (B. 5- pr- lO' 

and /. O = 4 (B. 5, pr. 9). 

Thirdly, if ^ 13 , then will O ZI ♦ ; 
becaufe C W ^"d : ^ = ^ : Q ; 

/. ^ C O, by the firft cafe, 
that is, Q 13 ^ . 

/. If the firft has the fame ratio, &c. 



BOOKV. PROP. XV. THEOR. 179 




AGNITUDES /lave the fame ratio to one another 
which their equimultiples have. 



Let and be two magnitudes ; 
then, : ■ : : M' : M' ^ ^^ 

For A : = a 



.*. # : H :: 4 • : 4 • (B. 5- pr- 12)- 

And as the fame reafoning is generally applicable, we have 
# : ■ :: M' A : M'h. 



/, Magnitudes have the fame ratio, &c. 



i8o BOOKF. DEFINITION XIII. 



DEFINITION XIII. 

The technical term permutando, or alternando, by permu- 
tation or alternately, is ufed when there are four propor- 
tionals, and it is inferred that the firft has the fame ratio to 
the third which the fecond has to the fourth ; or that the 
firft is to the third as the fecond is to the fourth : as is 
Ihown in the following propofition : — 

Let# : 4 ::19 :B) 

by " permutando" or "alternando" it is 
inferred ^ : ^ •• ^ • B • 

It may be neceffary here to remark that the magnitudes 
A, ^j V7H7 muft be homogeneous, that is, of the 
fame nature or fimilitude of kind ; we muft therefore, in 
fuch cafes, compare lines with lines, furfaces with furfaces, 
folids with folids, &c. Hence the ftudent will readily 
perceive that a line and a furface, a furface and a folid, or 
other heterogenous magnitudes, can never ftand in the re- 
lation of antecedent and confequent. 



BOOK V. PROP. XVL THEOR. 



i8i 




F four magnitudes of the fame kind be proportionals, 
they are afo proportionals ivhen taken alternately. 



Let ^ : Q : : H : ▲ , then ip : B - U • ^ • 

ForM fl : M O :: ^ : Q (B. 5. pr. 15), 

d M ^ : M Q :: H : ^ (^yP-) ^nd (B. 5. pr. 11) 

alfo /;; m : /;; ▲ ' • H * ^ (^- 5- P''- ^ S) > 

.*. M ^ : M Q :: w : /« ^ (B. 5. pr. 14), 

and /. if M ^ C. =» or ^ zw B ? 

then will M Q C :=, or 33 ;« ^ (B. 5. pr. 14) ; 

therefore, by the fifth definition, 

.*. If four magnitudes of the fame kind, &c. 



1 82 BOOK F. DEFIXmOX XFL 



DEFLS'ITIOX XVI. 

DnmxxDO, by di^ i :- . r - :h ere ire :": _ : r : : : ; - r , 

and it is inferred, l i : J-.e exceli : : : - : toood 

b to the fecood, £i iJie ev; ;::;::::; :r : -, 
b to tbe fenrth. 

le: : 3 ::C : D; 

far ** diridendo ** it b inferred 

A miners B : B : : C minns '^ : ~" . 

Ac; ; r :: . r :: -e,A b fbppt^i :: r-e rti ;' ^ : 
B, and C i:'- -" ; if thb be -:: ±: :i : :ut to 

have r : i :: ::..- :£ D greater iIjj: .2 : 



S :A :-. D :C; 
-A :A :: zuz^C :C. 



BOOK V. PROP. XVII. THEOR. 183 




[F magnitudes, taken jointly, be proportionals, they 
Jhall alfo be proportionals ii-hen taken feparately : 
\, that is, if tivo magnitudes together have to one of 
them the fame ratio which two others have to one 
ofthefe, the remaining one of the fir ft two Jhall have to the other 
the fame ratio which the remaining one of the laft two has to the 
other of thefe. 

Let tp + CI: O ::" + ♦: ♦, 
then will ^ : O :: ■ : ♦. 

Take M ^ C « O to each add M Q, 

then we have M V + M Q C 'w O + M Q, 

orM(V + CI) C (^^ + M: D: 

but becaufe IP + 0:0::"+#: ♦ (hyp.), 

and M (IP + O) C (;« + xM) Q ; 

.-. M (■ + ♦) C (^^ + M) 4 (B. 5. def. 5) ; 

/. M ^ + M ♦[=//;♦+ M ♦ ; 

.'. M '^ C ^ ^ . by taking M ^ from both fides : 

that is, when 'SI ^ ^ m U, then M T~ m ^ . 

In the Tame manner it may be proved, that if 
M ^ r= or ^ OT U, then will M =r or — \ m ^ • 
and /. V : O : : ? : ♦ (B. 5. def. 5). 

.*. If magnitudes taken jointly, &c. 



l84 book V. DEFINITION XV. 



DEFINITION XV. 

The term componendo, by compofition, is ufed when there 
are four proportionals ; and it is inferred that the firft toge- 
ther with the fecond is to the fecond as the third together 
with the fourth is to the fourth. 

Let A : B : : : D ; 

then, by the term " componendo," it is inferred that 
A-|.B:B:: -j-D:D. 

By " invertion" B and O may become the firft and third, 
A and _ the fecond and fourth, as 

B : A : : D : C , 

then, by " componendo," we infer that 
B + A : A ; : D -|- . : ^ . 



BOOK F. PROP. XVIII. THEOR. 



i8s 



F magnitudes, taken feparately, be proportionals , 
they fliall alfo be proportionals when taken jointly : 
that is, if the Jirji be to the fecond as the third is 
to the fourth, the firji. and fecond together fhall be 
to the fecond as the third and fourth together is to the fourth. 




Let IP : O 
then fP + Q : Q 
for if not, let |p -f- Q 
fuppofing ^ 

• • ^^ • v^ • • 

but ^ : Q : : 



not = ^ ; 

• (B. 5. pr. 17); 

: ^ (hyp.); 



.'•■:#::■: 4 (B. 5. pr. n); 

•••• = ♦ (B. 5- pr- 9). 

which is contrary to the fuppofition ; 

.'. ^ is not unequal to ^ ; 
that is =: ^ ; 

*, If magnitudes, taken feparately, &c. 



B B 



i86 



BOOK V. PROP. XIX. THEOR. 




F a isohole magnitude be to a whole, as a magnitude 
taken from the firji, is to a magnitude taken from 
the other ; the remainder ffoall be to the remainder, 
as the ivhole to the whole. 



Let l^ + O :■ + ♦:: IP :■, 
then will Q: ::'P> + 0:H+'', 

For tP + a : V :: ■ + t : ■ (^l^er.), 

.*. O : V ••: ♦ :■ (divid.), 

again Q : 4 ^^ 9 ^ H (alter.), 

butlP + 0:» + # ::^:B hyp.); 

therefore Q : : : ^ + D : ■ + ♦ 

(B. 5. pr. 11). 

,*, If a whole magnitude be to a whole, &c. 



DEFINITION XVII. 

The term " convertendo," by converfion, is made ufe of 
by geometricians, when there are four proportionals, and 
it is inferred, that the firft is to its excefs above the fecond, 
as the third is to its excefs above the fourth. See the fol- 
lowing propofition : — 



BOOK V. PROP. E. THEOR. 



187 




F four magnitudes be proportionals, they are alfo 
proportionals by converjion : that is, the Jirjl is to 
its excefs above the fecond, as the third to its ex- 
cefs above the fourth. 

then fhall • O • • ^ • ■ '-W, 



Becaufe 
therefore ' 

.-. o 

•. #0: 



:0::B : (divid.), 

i :: ^ : ■ (inver.). 



(compo.). 



.'. If four magnitudes, &c. 



DEFINITION XVIII. 

" Ex squall " (fc. diflantia), or ex aequo, from equality of 
diftance : when there is any number of magnitudes more 
than two, and as many others, fuch that they are propor- 
tionals when taken two and two of each rank, and it is 
inferred that the firft is to the laft of the firft rank of mag- 
nitudes, as the firft is to the laft of the others : " of this 
there are the two following kinds, which arife from the 
different order in which the magnitudes are taken, two 
and two." 



i88 BOOK V. DEFINITION XIX. 



DEFINITION XIX. 

" Ex asquali," from equality. This term is ufed iimply by 
itfelf, when the firft magnitude is to the fecond of the firft 
rank, as the firft to the fecond of the other rank. ; and as 
the fecond is to the third of the firft rank, fo is the fecond 
to the third of the other ; and fo on in order : and the in- 
ference is as mentioned in the preceding definition; whence 
this is called ordinate proportion. It is demonftrated in 
Book 5. pr. 22. 

Thus, if there be two ranks of magnitudes, 

A, B, , . , E, F, the firft rank, 

and L, M, N , < ' , P, Q, the fecond, 

fuch that A : B : : L : M, B : :: M : , 

C : U : : .\ : ( ) , D : E : : o : P, E : F : : P : Q ; 

we infer by the term " ex squali" that 

A : F :: L :Q. 



BOOK F. DEFINITION XX. 189 



DEFINITION XX. 

" Ex ^quali in proportione perturbata feu inordinata," 
from equality in perturbate, or diforderly proportion. This 
term is ufed when the firft magnitude is to the fecond of 
the firft rank as the laft but one is to the laft of the fecond 
rank ; and as the fecond is to the third of the firft rank, fo 
is the laft but two to the laft but one of the fecond rank ; 
and as the third is to the fourth of the firft rank, fo is the 
third from the laft to the laft but two of the fecond rank ; 
and fo on in a crofs order : and the inference is in the i8th 
definition. It is demonftrated in B. 5. pr. 23. 

Thus, if there be two ranks of magnitudes, 
A. , B , C , D , E , F , the firft rank, 
and , M , N , O , P , Q , the fecond, 
fuch that A : B : : P : Q , B : C : : O : P , 
C^ : D : : N : O , D : ' : : * : N , : : : : vr ; 
the term " ex xquali in proportione perturbata feu inordi- 
nata" infers that 
A : r : : ^ : <,> . 



190 



BOOK V. PROP. XX. THEOR. 




F i/iere be three magnitudes , and other three, which, 
taken two and two, have the fame ratio ; then, if 
the jirjl be greater than the third, the fourth fiall 
be greater than the fixth ; and if equal, equal ; 
and if lefs, lefs. 

Let ^, 0> J be the firft three magnitudes, 
and ^, Oj ^> be the other three, 

fuch that fp :0 ::4 :0,andC) :B ::0:#- 

Then, if ^ IZ> =» or Z] , then will ^ CI, =, 

orZl ^. 
From the hypothefis, by alternando, we have 

andO :0 ::■:•; 

.*. "P :♦::■: • (B. 5- pr- n); 

/. if I^F d, =, or Z] , then will ^ C =, 

orI3 (B. 5. pr. 14). 

,*, If there be three magnitudes, 6cc. 



BOOK V. PROP. XXL THEOR. 



191 



F t/iere be three magnitudes, arid other three which 
have the fame ratio, taken two and two, but in a 
crofs order ; then if the fir ft magnitude be greater 
than the third, the fourth fliall be greater than the 
fixth ; and if equal, equal ; and if lefs, lefs. 




Let 



I, be the firft three magnitudes, 



and ^, O*, fpt, the other three, 

fuch that ^ : A : : O •# > ^"^ A ' H - • ^ ■ O ' 

Then, if f C. =. or ID ■, then 
will ♦ C =, Zl #. 

Firft, let ^ be C ■ : 

then, becaufe ^ is any other magnitude, 

¥•*!=■' A (6. 5-pr.8); 

butO :#::¥: A (^yp-); 
.-. O :# !=■ :A (B. 5-pr- 13); 

and becaufe {^ : ■ :: ^ : (j (hyp.) ; 
and it was fliown that (^ '. % d H ' iil > 

.*. O : " C C : ♦ (B. 5- pr- 13); 



192 BOOK F. PROP. XXI. THEOR. 

•• • =] ♦, 

that is ^ C I . 

Secondly, let ^ H ; then fhall ^ = ^. 

For becaufe ^ B, 

V :* = ■ :dl (B. 5.pr.7); 

but : il = 0> : (hyp.). 

and ^ * A = O : ^ (hyp- ^"'l ifiv.), 

.-. O : # = : ♦ (B. 5. pr. II), 

.-. ^ = i (B. 5. pr. 9). 

Next, let be Z3 ■? then ^ fhall be Z3 ; 

for B C 
and it has been (hown that (§ • ^ ^ ^ * ▼' 

and il : = : O; 

/. by the firft cafe is C ^j 
that is, ^ ^ 9 . 

/. If there be three, &c. 



BOOK V. PROP. XXII. THEOR. 



193 



F there be any number of magnitudes, and as nuuiy 
others, 'which, taken two and two in order, have 
the fame ratio ; the frji JJjall have to the lajl of 
the firft magnitudes the fame ratio which the frji 
of the others has to the lajl of the fame . 

N.B. — This is ifually cited by the words "ex (egua/i," or 
"ex cequo." 




irft, let there 


36 magnitud 


es^ 




and as many others ▲ 


,0 = 


? 




fuch that 






w •• 


♦ "♦ : 


0, 




and ^ 


:il ::0 




• 

> 


then fliall 


1^ • ^ • • 
▼ • 


♦ = 


■«. 



Let thefe magnitudes, as well as any equimultiples 
whatever of the antecedents and confequents of the ratios, 
lland as follows : — 



and 

M ^,« ♦, N ' , M ^, w <;>, N 1, 

becaufe |p : ^ : : ^ : O ? 

:: M ^ :/«<3 (B. 5. p. 4). 

For the fame reafon 

w ^ : N : ; /« <^ : N | ; 

and becaufe there are three magnitudes, 
c c 



/.Mm: m 



194 BOOK F. PROP. XXII. THEOR. 

and other three, M ^ , w <^ , N , 
which, taken two and two, have the fame ratio ; 

/. ifMip CZ, =, ori:N B 

then will M ^ CZ. =. oi' Z] N , by (B. 5. pr. 20) ; 

and .*. ^ : ■ : : ^ : # (def. 5). 

Next, let there be four magnitudes, ^ , ^, H ^ ^ » 

and other four, ^ , ^, IB , ▲ , 

which, taken two and two, have the fame ratio, 

that is to fay, ^ • ^ • • O ' #' 

♦ :■::•: , 

and A : ^ ::m : ▲, 

then fhall IP : ^ : : O * ^ ' 

for, becaufe l[p , ^^, , are tliree magnitudes, 

and <^ , ^f , other three, 

which, taken two and two, have the fame ratio ; 

therefore, by the foregoing cafe, ^^ : ■ : : (2> • ^, 

but a : 4 : • «■ : -^ ; 

therefore again, by the firfl; cafe, ip : ^ : : (^ '- ^ f 

and fo on, whatever the number of magnitudes be. 

,*, If there be any number, &c. 



BOOK V. PROP. XXIII. THEOR. 



195 




F t/iere be afiy number of tnagnitudes, and as many 

others, ivhich, taken two and two in a crofs order, 

have the fame ratio ; the firjl fliall have to the laji 

of the firjl magnitudes the fame ratio which the 

firji of the others has to the laji of the fame. 

N.B. — This is ifually cited by the words " ^x aquali in 
proportione perturbatd ;" or " ex aquo perturbato." 

Firft, let there be three magnitudes, ^j(^> |) 

and other three, ' > O ' ^ » 

which, taken two and two in a crofs order, 

have the fame ratio ; 



o 



Let thefe magnitudes and their refpective equimuhiples 
be arranged as follows : — 

M ,M^,m^,M ,,,m(^,m%, 

then f IQ ::M ' : M Q (B. 5. pr. 15); 

and for the fame reafon 

but^ :q ::<2> :0 (hyp.). 



that is, |; : 


U ' 


:o 


and Q 


:■ : 


•♦ 


then fhall ^ 


:■ : 


= ♦ 



196 



BOOK V. PROP. XXIII. THEOR. 



.-. M ip :MQ ::<^ :# (B. 5. pr. n); 

and becaufe O : ■ : : ^ : <2> (Jiyp-)> 

.-. M Q : w H : : ^ : w ^ (B. 5. pr. 4) ; 

then, becaufe there are three magnitudes, 

M W, M Q, w ■, 

and other three, M , m (2), w ^ , 

which, taken two and two in a crofs order, have 

the fame ratio ; 

therefore, if M [^, ^, or "H ;;; J j 

then will M [Z, =r, or ;i] ;;/ (B. 5. pr. 21), 

and /. ,;; : ■ :: -J. : # (B. 5. def. 5). 

Next, let there be four magnitudes, 

and other four, (2)j ^j ■> A.? 

which, when taken two and two in a crofs order, have 
the fame ratio ; namely. 



IP 


:D 


:: ■ 


D 


■ 


::• 


andH 


• •# 


-0 


en fhall 




"O 



For, becaufe ^^ ^, | are three magnitudes, 



BOOKF. PROP. XXIII. THEOR. 197 

and 9, SI, i^, other three, 

which, taken two and two in a crofs order, have 

the fiime ratio, 
therefore, by the firfl cafe, ^ : H •• • ^^ 

but ■ : :: <^ : #, 

therefore again, by the firft cafe, y : ^ : : /S ' A ? 
and i'o on, whatever be the number of fuch magnitudes. 

.*. If there be any number, &c. 



198 



BOOK V. PROP. XXIV. THEOR. 



jF the firji has to the fecond the fame ratio which 
the third has to the fourth, and the fifth to the 
fecond the fame which the fix th has to the fourth, 
the fir fi and fifth together Jhall have to the fecond 

the fame ratio which the third and fix th together have to the 

fourth. 




First. 



Fifth. 



Second. 

D 



Third. 



Sixth. 



Fourth. 



Let ip : 


U: 


:a:<^, 


and (2> : 


D: 


:•:#. 


'+0 


•Q: 


• ■ + • : 4 



then 

For <2>:D--: #: ^ (%P-). 

and Q : ^ :: ^ : B (^yP-) ^"^ (invert.), 

.-. 0> •¥::#:■ (B- 5- Pr- 22); 

and, becaufe thefe magnitudes are proportionals, they are 

proportionals when taken jointly, 

.•• V+ 0:0:: •+ ■: • (B. 5- pr- 18), 

but o : D • : • • '- (hypO. 

.-. V + O : U ::#+■• t (B. 5- pr. 22). 



/. If the firft, &c. 



BOOK V. PROP. XXV. THEOR. 



199 




F four magnitudes of the fame kind are propor- 
tionals, the greatejl and leaf of them together are 
greater than the other two together. 



Let four magnitudes, ■ -j- ^, H -|- ■-' , |^, and |^ , 
of the fame kind, be proportionals, that is to fay, 

and let ■ -f- O ^^ ^^ greateft of the four, and confe- 

quently by pr. A and 14 of Book 5, ^ is the leaft ; 

then will ^+1314- beClB+ +D; 

becaufe If + Q :■+>:: O : ♦, 



but 



+ Dl= ■ + 



(B. 5. pr. 19), 

(hyp.). 



.'. "f [= ■(B. 5. pr. A); 
to each of thefe add O "4" ^7 

•*. fP + O + 1= ■ + o + ♦■ 

If four magnitudes, &c. 



2o,o BOOK V. DEFINITION X. 



DEFINITION X. 

When three magnitudes are proportionals, the firfl is laid 
to have to the third the dupHcate ratio of that which it has 
to the fecond. 

For example, if A, b', C, be continued proportionals, 
that is, A : B :: B : C, A is faid to have to C the dupli- 
cate ratio of x\ : B ; 

or — r= the fquare of — . 
This property will be more readily feen of the quantities 

'J ^"f , , J, tor /T !' '. u ' '.'. li ■ '• a \ 

and — ^ r^ r= the fquare of — = r. 



or of iJy 



f jr~ , 



for — ^ -3 = the fquare of — =:— . 
a r " ' 



DEFINITION XI. 

When four magnitudes are continual proportionals, the 
firft is faid to have to the fourth the triplicate ratio of that 
which it has to the fecond ; and fo on, quadruplicate, &c. 
increafing the denomination ftill by unity, in any number 
of proportionals. 

For example, let. A, B, C, D, be four continued propor- 
tionals, that is, A ; : : : : C :: C : D ; A is faid to have 
to D, the triplicate ratio of N to iJ ; 

or - := the cube of—. 



BOOK K DEFINITION XL 201 

This definition will be better underftood, and applied to 
a greater number of magnitudes than four that are con- 
tinued proportionals, as follows : — 

Let^r", ' yar> ^y be four magnitudes in continued pro- 
portion, that is, ^ »■':': : '■ ar '-'-ar '• (i, 

. ar' „ , , -ar^ 

then =: r" r= the cube or — ^ r. 

a 

Or, let ar', ar*, ar^, ur', ar, a, be fix magnitudes in pro- 
portion, that is 

ar* : rtr* :: ar^ ■ ar* :: ar" : ar" :: ar' : ar :: ar : a, 

a r - a r 

then the ratio — = r" zrz the fifth power of — : zr: r. 
a ^ rtr* 

Or, let a, ar, ar^, ar^, ar*, be five magnitudes in continued 
proportion; then — 5 := -5 =z the fourth power of — ::=:-. 



DEFINITION A. 

To know a compound ratio : — 

When there are any number of magnitudes of the fame 
kind, the firfi: is faid to have to the lafl: of them the ratio 
compounded of the ratio which the firfl has to the fecond, 
and of the ratio which the fecond has to the third, and of 
the ratio which the third has to the fourth ; and fo on, unto 
the lafl; magnitude. 

For example, if A , B , C , D , 
be four magnitudes of the fame 
kind, the firft A is faid to have to 
the lafl: D the ratio compounded 
of the ratio of A to B , and of the 
ratio of B to C , and of the ratio of C to D ; or, the ratio of 

DD 





A 


B 


C 


D 




E 


F 


G 


H 

s 


K 


L 



202 BOOKF. DEFINITION A. 

A to D is faid to be compounded of the ratios of \ to B , 
B to C , and c to |j. 

And if A has to B the fame ratio which 1 has to V , and 
B to C the fame ratio that G has to H, and C to D the 
fame that K has to L ; then by this definition, \ is said to 
have to L> the ratio compounded of ratios which are the 
fame with the ratios of E to F, G to H, and K to L. And 
the fame thing is to be underftood when it is more briefly 
exprefled by faying, \ has to D the ratio compounded of 
the ratios oft to F, G to H, and K to I . 

In like manner, the fame things being fuppofed ; if 
has to the fame ratio which \ has to D, then for fhort- 
nefs fake, is faid to have to the ratio compounded of 
the ratios of E to F, G to H, and K to L. 

This definition may be better underftood from an arith- 
metical or algebraical illuftration ; for, in fact, a ratio com- 
pounded of feveral other ratios, is nothing more than a 
ratio which has for its antecedent the continued produdl of 
all the antecedents of the ratios compounded, and for its 
confequent the continued produdl of all the confequents of 
the ratios compounded. 

Thus, the ratio compounded of the ratios of 

2 : ;, 4 : 7, 6 : 1 1, 2 : 5, 

is the ratio of ; X X 6 X 2 : X X 1 1 X 5, 

or the ratio of 96 : 11 55, or -^2 : 385. 

And of the magnitudes A, B, C, D, E, F, of the fame 
kind, A : F is the ratio compounded of the ratios of 
A : B, B : C C : D, D : E, E : F ; 
for A X B X X X E : B X C X x E X F, 

^^ nx'x XEXF = T' ""^ ^^^ ""^"^ °^ "^ '■ ^' 



BOOK r. PROP. F. THEOR. 



203 




ATIOS wAic/i are cojnpounded of the fame ratios 
are the fame to one another. 



Let A : B : : F : G, 
B : C :: G : H, 
C: D::H:K, 

and D : E :: K : L. 



A B C D E 
F G H K L 



Then the ratio which is compounded of the ratios of 
A : R, ^ : , : , : t , or the ratio of A : E, is the 
fame as the ratio compounded of the ratios of F : G, 
G : H, H : K, K : L, or the ratio of F : L. 





For ^ = 


F 

G' 








B 

C ~" 


G 
H' 








C __ 
D "■ 


H 
K' 








a„d^ = 


K 






AX 


XX 


F X 
X 


X 
X 


X -: 


X 


X X ■ — 


X L 




and /. - 


F 
— L 







or the ratio of A : E is the fame as the ratio of F : L. 

The fame may be demonflrated of any number of ratios 
fo circumftanced. 



Next, let A : B : : K : L, 
B: C:: H: K, 
C: D:: G: H, 
D: E :: F: G. 



204 BOOK V. PROP. F. THEOR. 

Then the ratio which is compounded of the ratios of 
A : B, B : C, C : D, D : E, or the ratio of A : E, is the 
fame as the ratio compounded of the ratios of :L, : K, 
G : H, F : , or the ratio of F :L. 



For - = -, 



I 



and — =: — ; 

r. 
A X X X D . X X X F 



X ^^ X E — L X X X G * 



^•■"^ •••! = -' 



F 

L 

or the ratio of A : ¥ is the fame as the ratio of F : L. 
,", Ratios which are compounded, &c. 



BOOK V. PROP. G. THEOR. 



205 



F fever al ratios be the fame to fever al ratios, each 

to each, the ratio which is compounded of ratios 

which are the fame to the firft ratios, each to each, 

jhall be the fame to the ratio compounded of ratios 

which are the fame to the other ratios, each to each. 




A B C: D E ¥ G H 


P Q R S T 


a bed e f g h 


V w X y 



If A : B : : d : ^ 


and A : B : : P : 


Q 


a:b:: 




: \\ 


CD ::€ -.d 


C:D::Q: 


R 


c:d:: 


w 


: X 


E:F ::e:f 


E:F ::R 


S 


e:f:: 


X 


: Y 


and G : II :: g : A 


G:H:: S : 


T 


g:h:: 


Y 


: Z 


then P : T = ^ " 


• • 








p^^ P A a 


Z3 


> 






2 — ^' - i- 

R D d 


= 


> 






R __ E e 

S" — * F — 7 


^ 


9 






^ G ff 










f H h 




) 






and • '' X 9 X k X ■ __ 
^""^ • • X R X s X r — 


X 
X 


X X 
X X 


~ » 




and /. -p = - 


~ > 








01 


rP : T = : 


'/.. 









If feveral ratios, &c. 



2o6 



BOOK V. PROP. H. THEOR. 




F a ratio which is compounded of fever al ratios be 
the fame to a ratio which is compounded of fever al 
other ratios ; and if one of the firjl ratios, or the 
ratio which is compounded of fever al of them, be 
the fame to one of the laji ratios, or to the ratio which is com- 
pounded of Jeveral of them ; then the remaining ratio ofthefirjl, 
or, if there be more than one, the ratio compounded of the re- 
maining ratios, JJi all be the fame to the remaining ratio of the 
la/i, or, if there be more than one, to the ratio compounded of thefe 
remaining ratios. 



A 


B 


C 


D 


E 


F 


G 


H 




F 


Q 


R 


S 


T 


X 





Let A : B, B : C, C : D, D : E, E : F, F : G, G : H, 
be the firft ratios, and P : Q^_Qj^R, R : S, S : T, T : X, 
the other ratios ; alfo, let A : H, which is compounded of 
the iirfl: ratios, be the fame as the ratio of P : X, which is 
the ratio compounded of the other ratios ; and, let the 
ratio of A : E, which is compounded of the ratios of A : B, 
B : C, C : D, D : E, be the fame as the ratio of P : R, 
which is compounded of the ratios P : Q,^ Qj R. 

Then the ratio which is compounded of the remaining 
firft ratios, that is, the ratio compounded of the ratios 
E : F, F : G, G : H, that is, the ratio of E : H, fhall be 
the fame as the ratio of R : X, which is compounded of 
the ratios of R : S, S : T, T : X, the remaining other 
ratios. 



Becaufe - 



BOOK V. PROP. H. THEOR. 207 



, X f. X C X D X L X J: X & P X Q X R X S X I 



l; X L X D X E X F X G X H Q X R X s X 'I' X X' 

, X [; X C X 1^ w K X F X G P X Q w R X S x f 

f! X L X D X K •^ h- X(,XU — O X R ^ ^ X T X X' 

anH ^ X B X C X D _ P X Q 
^ E XCX DX E — Qx R' 

, E X F X G R X - X I 

• * F X G X H i X I X X» 

/. E : H = R : X. 
,*, If a ratio which, &c. 



2o8 



BOOK V. PROP. K. THEOR. 




F t/iere be any number of ratios, and any number of 
other ratios, fuch that the ratio which is com- 
pounded of ratios, which are the fame to the frji 
ratios, each to each, is the fame to the ratio which 
is compounded of ratios, which are the fame, each to each, to 
the lajl ratios — and if one of the firji ratios, or the ratio which 
is compounded of ratios, which are the fame to federal of the 
firjl ratios, each to each, be the fame to one of the lajl ratios, 
or to the ratio which is compounded of ratios, which are the 
fame, each to each, to fever al of the lajl ratios — then the re- 
maining ratio of the firjl ; or, if there be more than one, the 
ratio which is compounded of ratios, which are the fame, each 
to each, to the remaining ratios of the firJi, Jhall be the fame 
to the remaining ratio of the lajl ; or, if there be more than 
one, to the ratio which is compounded of ratios, which are the 
fame, each to each, to thefe remaining ratios. 



h k m n s 




AB, CD, EF, GH, K L, MN. 


a b c d e t g 


O P , O R , S T , V W , X Y , 


h k I m n p 


abed e i g 





Let A:B, C:D, E:F, G:H, K:L, M:N, be the 
firft ratios, and o :!', (^:R , ^ :T, V :W, X : , the 
other ratios ; 



and let A : B 


zn a '. b , 


C :D 


= b :c. 


E :F 


HZ L id. 


G :H 


~~7. 'i : '' J 


K : L 


— e :f. 


M:N 


T"^ / I P" • 



BOOK F. PROP. K. THEOR. 209 

Then, by the definition of a compound ratio, the ratio 
of ,7 In- is compounded of the ratios of j :/,, /; ic, c -Jt J ','> 
g :/"./"■?-. which are the fame as the ratio of A : B, C : D, 
E : F, G : H, K : L, M : N, each to each. 



Alfo, 


:^ ^; h 


•k. 


Q^ 


\R •=. k 


:/. 


: : 


T = /: 


m. 


V : 


VV = m 


: n. 


-- ; 


— n 


P- 



Then will the ratio oi h\p be the ratio compounded of 
the ratios of h:k, k:l, I '.my m'.n, n:p, which are the 
fame as the ratios of :p , Ct :R , S :T , V :W , X :Y , 
each to each. 

/, by the hypothefis a '•!? = h:p. 

Alfo, let the ratio which is compounded of the ratios of 
A: B, C : D, two of the firfl: ratios (or the ratios of j ict 
for \ : ^ = J : A, and C" : P = , : ), be the fame as the 
ratio of a : d, which is compounded of the ratios of a : b, 
b : c, c : d, which are the fame as the ratios of : , 
: , : , three of the other ratios. 

And let the ratios of h : s, which is compounded of the 
ratios of h : k, k : m, m : n, n : s, which are the fame as 
the remaining firft ratios, namely, E : F, G : H, K : L, 
M : N ; alfo, let the ratio of e : g, be that which is com- 
pounded of the ratios e : f, f : g, which are the fame, each 
to each, to the remaining other ratios, namely, V :W, 
X : Y . Then the ratio of h : s fhall be the fame as the 
ratio of e : g ; or h : s r= erg. 

p AXCXKX'.XKXM g X 6 X ,- X ,i X r X /• 

:-^ X 11 X F X H X I X X — - 6 X c X i X . X/ X y ' 

£ £ 



2IO BOOK V. PROP. K. THEOR. 

, OX ox ^X . X ■ ^x^x^x?»x» 

^ X R X I X V, X kXlXmXnXp 

by the compolition of the ratios ; 

. cX/XcXrfXfX; h X kX I XrnXn 



''iXcXdXfXfXj kX I Xm X n Xp 



(hyp.). 



uXl w c X ^ X £ X / kX kX I w 

.7 X c ^ dX cX/Xg kX t Xm ^ n Xp' 

but — — A X C __ ^- X X . a Xb Xc __ h Xk X i . 

;X. £XD ;X X bxcxd kX I Xm' 

, f X -J X f X ; _. »t X n 
' ' i X e X fX i' n Xp' 



A„J C X ^" X t X • h X k X m X n ,, . 

And .v. V^vV — k VmVn Vs ("7?-). 



>?Xc X/X^ 




k 


X m 


X 


n X s 


and 


m X n 
n Xp 


— 


e 
T 


X f 

Xg 


(hyp.). 


• 
• • 


h X k X m 
k X m X n 


X 
X 


n . 

s 


— 


e f 




• 
• • 


s 


^^ 


e 








/. h 


: s : 


— 


e : 


g- 





/, If there be any number, &c. 



; * Al"-ebraical and Arithmetical expositions of the Fifth Book of Euclid are given m 
Hyrne's Doctrine of Proportion ; published by Williams and Co. London. 1841. 









BOOK VI. 
DEFINITIONS. 

I. 

ECTILINEAR 

figures are faid to 

be fimilar, when 

they have their fe- 
veral angles equal, each to each, 
and the fides about the equal 
angles proportional. 

II. 

Two fides of one figure are faid to be reciprocally propor- 
tional to two fides of another figure when one of the fides 
of the firft is to the fecond, as the remaining fide of the 
fecond is to the remaining fide of the firft. 

III. 

A STRAIGHT line is faid to be cut in extreme and mean 
ratio, when the whole is to the greater fegment, as the 
greater fegment is to the lefs. 

IV. 

The altitude of any figure is the straight line drawn from 
its vertex perpendicular to its bafe, or the bafe produced. 



2;2 



BOOK VI. PROP. I. THEOR. 





RIANGLES 

and parallelo- 
grams having the 
fame altitude are 
to one another as their bafes. 



Let the triangles 



1 and m 



have a common vertex, and 



their bafes 



and 



in the fame ftraight hne. 



Produce i both ways, take fucceffively on 

— — produced lines equal to it ; and on — — — pro- 
duced lines succefTively equal to it ; and draw lines from 
the common vertex to their extremities. 



A 



The triangles j^-JKJt^ thus formed are all equal 
to one another, fmce their bafes are equal. (B. i . pr. 38.) 



A 



and its bafe are refpectively equi- 



i 



multiples of ■ and the bafe 



BOOK VL PROP. I. THEOR. 2- 3 



^ 



In like manner » _ and its bafe are refpec- 



i 



lively equimultiples of |^ and the bafe — — . 

.*. Ifm or 6 times ^ (^ := or 13 « or 5 times B 
then m or 6 times — — C ^ or ;^ « or 5 times u-m i , 
m and « ftand for every multiple taken as in the fifth 
definition of the Fifth Book. Although we have only 
fhown that this property exifts when m equal 6, and n 
equal 5, yet it is evident that the property holds good for 
every multiple value that may be given to m, and to n. 



a 



(B. 5. def 5.) 



Parallelograms having the fame altitude are the doubles 
of the triangles, on their bafes, and are proportional to 
them (Part i), and hence their doubles, the parallelograms, 
are as their bafes. (B. 5. pr. 15.) 

Q. E. D. 



214 



BOOK VI. PROP. II. THEOR. 



* 





F a Jlraight line 




be draivn parallel to any 
Jide ■——■■-> of a tri- 
angle, it fliall cut the other 
tides, or thoj'e Jides produced, into pro- 
portional fegments . 

And if any Jlraight line ^— ^— 



divide the fides of a triangle, or thofe 
fides produced, into proportional feg- 
ments, it is parallel to the remaining 
fide —■•■-■■■■■», 



Let 



PART I. 



I, then {hall 



• •• ' ■•• 



Draw 



and 



and 




(B. I. pr. 37); 



■V- 



V 




\- 



^ : ! \ (B.5.pr.7);but 



(B. 6. pr. 1), 



(■■■■■■■■■* • I 



• «■■■■«■■■« * IttlllllBlB*, 



(B. 5. pr. II). 



BOOK VI. PROP. 11. THEOR. 



21 



PART II. 



Let 



Let the fame conftrudlion remain. 



becaulc 



: :: 1/ : • \ 



> (B.6. pr. I); 



and 



■ »«aiaflVB»« 




■ « 

• • • 



but 



«■■■ ^aai^ 




i\ 



/ 



-Z= 



(hyp.), 



), : i \ (B. 5. pr. 1 1 .) 



:. (B. 5- pr- 9) ; 

■■■■■■•. , and at the 



but they are on the fame bafe -■■■■■•■i 
fame fide of it, and 

•°. II (B- i.pr. 39) 



Q. E. D. 



2l6 



BOOK VI. PROP. III. THEOR. 




RIGHT /ine ( ) 

bifeSling the angle of a 
triangle, divides the op- 
pofite Jide into fegments 
— — ™) proportional 



to the conterminous Jides (- 

)• 

And if a Jiraight line (- 



— ) 
drawn from any angle of a triangle 
divide the oppojite Jide ( ■■■■■■) 

into fegments ( , ....■■■■..) 

proportional to the conterminous fides (— — — , ■ ), 

it bifeSls the angle. 



PART I. 



Draw -■ 



to meet 



then, ^ ^ (B. i.pr. 29), 

^ = < ; but ^ = 1 , .-. ^ =-# , 
.*. .......... = I I (B. I. pr. 6); 



and becaufe 



■ ■■■■■■■IHB 



(B. 6. pr. 2) ; 



but 



(B. 5. pr. 7). 



BOOK FI. PROP. III. THEOR. 217 



PART II. 

Let the fame conftrudtion remain, 



and 



(B. 6. pr. 2) ; 

but — ^.^ : ......... :: «-..—. ; ...^^ (hyp.) 



■■.■■■• 



(B. 5. pr. .1). 
and ,*, ■.■■•■■■a. zzz ■ (B. c. pr. o), 

and /. ^ = ^(B. I. pr. 5); but fince 
II «..; ^ _ ^^ 

and ^ = ^ (B. i. pr. 29); 
/. ^ =y, and =: ^, 



and .*. ........ bifedls ^ 



Q^E. D. 



F F 



«^i8 



BOOK VI. PROP. IV. THEOR. 





N equiatigular tri- 
angles ( ^ \ 



and ,-•'* \ ) the fides 
about the equal angles are pro- 
portional, and the Jides which are 
^L oppojite to the equal angles are 



homologous. 



Let the equiangular triangles be fo placed that two fides 

^^ and 



oppofite to equal angles 



^^^ may be conterminous, and in the fame ftraight line; 
and that the triangles lying at the fame fide of that flraight 
line, may have the equal angles not conterminous, 

ofite to jtKk , and fl^ to j^|^ . 



1. e. 



opp 



Draw -■•■■••■••t and 



', Then, becaufe 



▲ = ▲ 



II 



and for a like reafon, •■•■—■•■• 
- - „ / 



;B.i.pr.28); 

■ ""•"5 



is a parallelogram. 



But 



(B. 6. pr. 2) ; 



BOOK FI. PROP. IF. THEOR. 219 

and lince ' ^ — — — (B. i. pr. 34), 

^_i. : .......«i.. * and by 



• • 



alternation, — ^— ^ : — — — ^ j: -—.-««— ; — 

(B. 5. pr. 16). 

In like manner it may be fhown, that 

■^i^HMiMaBH^ ■«»•«■«■«»« ^^B^i^ia^^^ iiMiJiiaiiia* 

and by alternation, that 

•■■^^■^■Ma o^^^^^^^ JJ •■■■■■■•■^ J ■■■■■■■■■■* 

but it has been already proved that 

^a^^^^^^^m * ^MMHHWHHMM * ■■■■■■■■■■« 

and therefore, ex squali. 



■■■■■■■■■« * ■■■■■■■■■■■ 






(B. 5. pr. 22), 

therefore the fides about the equal angles are proportional, 

and thofe which are oppolite to the equal angles 

are homologous. 

Q. E. D. 



220 BOOK VI. PROP. V. THEOR. 





F tivo triangles have their Jides propor- 
tional (•■■•■•■■- : ■■«.■■■■» 

:: ^mmmmmmm, \ ) and 



(' 



■■■■■■■■■■■» * «•■■■■«■« 



:: — — ^— : i^^»— .) they are equiangular, 
and the equal angles are fubtended by the homolo- 
gous fides. 



From the extremities of 



, draw 



and 



, making 



W= iB (B. i.pr. 2.1;); 



= (B. I. pr. 32) , 

and fince the triangles are equiangular. 



(B. 6. pr. 4); 



but 



(hyp.); 



and confequently 



(B. 5. pr. 9). 



In the like manner it may be fhown that 



BOOK VI. PROP. V. THEOR. 221 

Therefore, the two triangles having a common bafe 
— «^— , and their fides equal, have alfo equal angles op- 



A =^a„d^ = ^ 



polite to equal fides, i. e. 

(B. I. pr. 8). 

But ^F = j^^ (conft.) 
and .*• jj^^ =: M/^ ^ for the fame 

reafon ^^^ z= flU? ^"^ 

confequently ^^ := (B. i. 32); 

and therefore the triangles are equiangular, and it is evi- 
dent that the homologous fides fubtend the equal angles. 

CUE. D. 



222 



BOOK VI. PROP. VI. THEOR. 




4 




.A 

F tivo triangles ( _^**^ _^*'^ 
and .^^___^ ) have one 

angle ( wKk ) o/' ///<' one, equal to one 

, angle ( m \ ) of the other, and the Jides 

^ about the equal angles proportional, the 

HPI^ triangles Jhall be equiangular, and have 

thofe angles equal which the homologous 

Jides fubt end. 



From the extremities of 



of 



Z:^ 



about 



^— , one of the fides 
■ \ , draw 



and 



maki 



in 



g 



▼ =A 



and ^W zz. 



^=4 



then ^ =: 



(B. I. pr. 32), and two triangles being equiangular. 



■ ■^■■■•■■■a 



(B. 6. pr. 4) ; 



but •••■>••••••• 



(hyp.); 



(B. 5. pr. 11), 



and confequently 



•«■■«■«•■•* 



(B. 5. pr. 9); 



k 



BOOK VI. PROP. VI. THEOR. 223 

.*. -^ \ = >,♦* in every refpedl. 

(B. I. pr. 4). 

But ^yf = j^ (conft.), 
and /. ZLj ■=. J^ ; and 

fince alio ■ \ ::z: JHl , 

^\ = -^ (B. i.pr. 32); 

j/\ 
and .*. A*I,....dW ^"d -^ \ are equiangular, with 

their equal angles oppolite to homologous Tides. 

Q^E. D. 



224. 



BOOK VI. PROP. VII. THEOR. 




A 



/\ 




F two triangles ( 



A 



and 



A 



» 



* * ) Aave one angle in 

• each equal ( ' equal to ^^ ), the 
\ Jides about two other angles proportional 

\ (-^ : — - :: : ..-—), 



^l 



A 



and each of the remaining angles ( 

and ^..^ ) either lefs or not lefs than a 
right angle, the triangles are equiangular, and thofe angles 
are equal about which the Jides are proportional. 

Firft let it be alTumed that the angles ^^ and <.^ 
are each lefs than a right angle : then if it be fuppofed 

that i^A ^"'^ ^^ contained by the proportional fides, 

are not equal, let ^\ be the greater, and make 

Becaufe ^ = ^ (hyp.), and ^\ = ^J (conft.) 
/. ^», = ^^--B ^B. I. pr. 32); 



\ 



BOOK VI. PROP. FIT. THEOR. 225 



(B. 6. pr. 4), 

but — ^^— : ■ :: — — ■■— : — .— .. (hyp.) 



• 9 

(B. 5. pr. 9), 



4 



and ,*. ^^ = ^^ (B. 1. pr. 5). 

But ^^B is lefs than a right angle (hyp.) 

,•, ^^^ is lefs than a right angle ; and ,', ^B muft 
be greater than a right angle (B. i. pr. 13), but it has been 

proved ^ '^^^.^ and therefore lefs than a right angle, 
which is abfurd. ,*. ^<^ and ^-\ are not unequal ; 
.', they are equal, and fince ^B rz / \ (hyp.) 



4=4 



(B. I. pr. 32), and therefore the tri- 
angles are equiangular. 



-^ and ^5 



But if '^^ and ^*-^ be aflumed to be each not lefs 
than a right angle, it may be proved as before, that the 
triangles are equiangular, and have the fides about the 
equal angles proportional. (B. 6. pr. 4). 

Q. E. D. 

OG 



226 



BOOK VI. PROP. Fin. THEOR. 





N a right angled 
triangle 







.j^S^ 9 ^^ 



( 

triangle and to each other. 
Becaufe ^^p» ~ 
common to 



i 
a perpendicular ( 

be drawn from the right angle 

to the oppojitejide, the triangles 



) on each Jide of it are fimilar to the whole 




(B. I. ax. 1 1 ), and 



and 




t\^< 



;B. I. pr. 32); 





and ..^^^^l are equiangular ; and 
conlequently have their Tides about the equal angles pro- 
portional (B. 6. pr. 4), and are therefore limilar (B. 6. 
def. I). 



In like manner it may be proved that ^^ is fimilar to 



k 




; but 




has been ihewn to be limilar 



to 





and 



k 



are 



fimilar to the whole and to each other. 



Q. E. D. 



BOOK VI. PROP. IX. PROB. 



22: 




ROM a given Jiraig/it line { " '" ) 
to cut off any required part . 

From either extremity of the 
given line draw — ^■—"••••■t. making any 

angle with ■ ; and produce 

■•••••> till the whole produced line 
■mtiBH* contains ■ > as often as 

-■■"■"■- contains the required part. 

Draw , and draw 



■ is the required part of 



For fi 



nee 




* ■«■■•••« 



(B. 6. pr. 2), and by compolition (B. 5. pr. 18) ; 

— ^— > -■".- mmmmmm 

but mmm 



'•■--•■ contains 



as often 



as 



contains the required part (conft.) ; 
■■■— is the required part. 



Q. E. D. 



228 



BOOK VI. PROP. X. PROB. 




and 



draw 




O divide a Jlraight 

line ( ) 

fanilarly to a 
given divided line 

)• 



From either extremity of 
the given line — i^ 

draw ■■■■««a>s»aaaKaj»«M 

making any angle ; take 



and 



>■•••«« equal to 



refpedlively (B. i. pr. 2) ; 



and draw — — --— and 

-— II to it. 



or 



and 



Since 



( — j are II, 



(B.6. pr. 2), 



(B. 6. pr. 2), 



and ,*, the given line 
fimilarlv to 



(conft.), 



(conft.). 



is divided 



Q.E. D. 



BOOK VL PROP. XL PROB. 



229 




O yf«i/ a third proportional 
to two given Jlraight lines 



At either extremity of the given 
line ^— i^— » draw .---—— 
making an angle ; take 
....... .^ = , and 

draw I : 



make .,.._... =: , 

and draw || — ^ • 

(B. I. pr. 31.) 
l u - is the third proportional 
to -■^— ^^ and _ 




For fince 



but 



(B. 6pr. 2); 



■(conft.) ; 



(B. 5. pr. 7). 



Q^E. D. 



230 



BOOK VI. PROP. XII. PROB. 




O find a fourth pro- 
portional to three 
given lines 



Draw 



and 

take 

and 

alfo 

draw 

and 



making any angle ; 



(B. r. pr. 31); 
is the fourth proportional. 



bU. (;■. 



On account of the parallels, 



(B. 6. pr. 2); 

•} = { = 



.} (conft.); 



■ ■■•■■■■•« • 



(B. 5. pr. 7). 



Q^E. D. 



BOOK VI. PROP. XIII. PROP. 



^31 




O Jind a mean propor- 
tional between two given 
Jlraight lines 



{ 



«ia«MMl«a«ni«l 



} 



Draw any ftraight line 
make — 




and 



bifed 



and from the point of bifedtion as a centre, and half the 



line as a radius, defcribe a femicircle 
draw — ^— ^— JL — — — 



cs 



is the mean proportional required. 



Draw 



and 




Since '^^^ is a right angle (B. 3. pr. 31), 
and ^^^^— is J_ from it upon the oppofite fide, 
•*. •^^"■^ is a mean proportional between 
— and ' (B. 6. pr. 8), 
and .*. between — — — and — •• (conft.). 



Q. E. D 



232 BOOK VI. PROP. XIV. THEOR. 





QJJ A L parallelograms 



\ 



and 




•which have one angle in each equal, 
have the Jides about the equal angles 
reciprocally proportional 



( 



■)• 



II. 



And parallelograms which have one angle in each equal, 
and the fides about them reciprocally proportional, are equal. 



Let 

and 



- and 



; and 



and 



^■^~", be fo placed that ' ■■■ ' ■ 
-■— may be continued right Unes. It is evi- 
dent that they mayaflume this pofition. (B. i. prs. 13, 14, 

1 5-) 



Complete 



% 



Since 



•V 




\ \ \ 



(B. 5. pr. 7.) 



BOOK VI. PROP. Xir. THEOR. 233 



(B. 6. pr. I.) 

The fame conftrudtion remaining : 
r 

A 








(B. 6. pr. I.) 

— (hyp.) 



(B. 6. pr. I.) 



(B. 5. pr. II.) 





and .*. ^Hi^ = ^^ (B. 5. pr. 9). 

Q^E. D. 



H H 



234 



BOOK VI. PROP. XV. THEOR. 




I. 

QUAL triangles, which have 
one angle in each equal 

( ^^ ^ ^B ), have the 

JiJt's about the equal angles reciprocally 
proportional 




(■ 



-- )• 



II. 



j^i 



And two triangles which have an angle of the one equal to 
an angle of the other, and the Jides about the equal angles reci- 
procally proportional, are equal. 



Let the triangles be {o placed that the equal angles 

^^ and ^A may be vertically oppolite, that is to lay, 

lb that ^mmmmi^m and — -^— may be in the lame 
ftraight line. Whence alfo i and -aiM^MMM mull 

be in the fame ftraight line. (B. i. pr. 14.) 

Draw ■— — — , then 



> 



(B. 6. pr. I.) 



(B. 5. pr. 7.) 



(B. 6. pr. I.) 



BOOK VI. PROP. XV. THEOR. 



235 



> 



(B. 5. pr. II.) 

II. 

Let the fame conftruction remain, and 



(B. 6. pr. I.) 



and 



A 



(B. 6. pr. I.) 

But — — . : ^— :: ; . ■ , (hyp.) 

(B.5 pr. 11); 
(B. 5. pr. 9.) 



• • • 



> -^ 



Q.E. D. 



236 



BOOK VI. PROP. XVI. THEOR. 



PART I. 

Y four Jh'ciight lines be proportional 



the reSlangle ( ■ 




: ■■). 

!■>..._.. ) contaified 
by the extremes, is equal to the rectangle 
X .........) contained by the means. 

PART II. 

And if the reSt- 
angle contained by 
the extremes be equal 
to the reBangle con- 
tained by the means, 
the four Jlraight lines 
are proportional. 



PART I. 
From the extremities of •— i» and " 

^M^BHB and ————— _L to them and ^ 



draw 



and ——•.-—- refpedlively : complete the parallelograms 
^^^^H and 



I 



And fince, 



• BB««»»«B» 


(hyp.) 




(conft.) 


• 


H (B. 6. pr. 14), 





BOOK VI. PROP. XVI. THEOR. 237 

that is, the redtangle contained by the extremes, equal to 
the redangle contained by the means. 

PART II. 
Let the fame conftrudlion remain ; becaufe 



• • 



(' 



and 11 ^. -....-..■. , 

(B. 6. pr. .4). 

But = , 

and — — i^ ^ — — — . ^conft.) 

(B. 5. pr. 7). 

Q. E. D. 



238 



BOOK VI. PROP. XVII. THEOR. 




fince 
then 




PART I 

F three Jlraight lines be pro- 
portional ( — 1 : ^^^mmt 

:: — — : ) the 

reSlangle under the extremes 
is equal to the fquare of the mean. 

PART II. 

And if the reSlangle under the ex- 
tremes be equal to the fquare of the mean, 
the three fir aight lines are proportional. 



PART I. 



Aflume 



X 



and 



X 



(B. 6. pr. 16). 



or 



But 

X 



■"9 

- X 



_.«i» ^ ; therefore, if the three ftraight hnes are 
proportional, the redlangle contained by the extremes is 
equal to the fquare of the mean. 





Aflume • 




PART II. 




« , then 

X 














• « 






• • 


and 


m» 9 




^^ • 


(B. 


6. pr. 16), 

• • 


wmm 





Q. E. D. 



BOOK VI. PROP. XVIIL THEOR. 239 




N a given Jlraight line ( ) ^,. 

to conftruSi a reBilinear figure 



fimilar to a given one ( 



and /imiiarly placed. 



^ 




Relblve the given figure into triangles by 
drawing the lines -«—--- and ••..••*•. 

At the extremities of — — — ^ make 

^ = Jb^^ and % = \J^ : 

again at the extremities of ■ 

and ^^ = ^^\ : in like manner make 

? = ^ ^"^ V = V • 




make -^^ =: 



Then 



-v 



is fimilar to 




It is evident from the conftrudlion and (B. 1. pr. 32) tliat 
the figures are equiangular ; and fince the triangles 



W ^" w 



are equiangular; then by (B. 6.pr.4), 



and 



240 BOOK VI. PROP. XVIII. THEOR. 
Again, becaule ^^^ and ^^B are equiangular. 



mm ** tt»afBffia«ai • 



/. ex asquali. 



(B. 6. pr. 22.) 

In like manner it may be fhown that the remaining fides 
of the two figures are proportional. 

.-. by (B. 6. def. i .) 
is fimilar to 





and fimilarly fituated ; and on the given line 

Q^E. D. 




BOOK VI. PROP. XIX. THEOR. 241 



I M I L A R trian- 



gles ( 



A 



and ^^^^^k ) are to one 
another in the duplicate ratio 
of their homologous Jides. 




Let 



^^ and A 



be equal angles, and 



and 



homologous fides of the fimilar triangles 



i^HHft and MKKKL 



and ^^^^^^ ^nd on -.-.-.—— the greater 
of thefe lines take --■— ■ a third proportional, fo that 



* «■■■■■■■■■ \ 



draw 



(B. 6. pr. 4) ; 



but 



(B. 5. pr. 16, alt.), 



■■■>■■■ 



MiSa 



(conll:.), 
— confe- 



1 1 



242 BOOK VI. PROP. XIX. THEOR. 



A\ 



quently ^^^ rz ^^ for they have the fides about 



the equal angles ^^ and ^Ik reciprocally proportional 

(B. 6. pr. 15); 

Aa-A\ 

(B. 5 pr. 7); 



^^^L : ^^ :: .... 

(B. 6. pr. I), 



that is to fay, the triangles are to one anotlier in tlie dupli- 
cate ratio of their homologous fides 
— i— and i^— -i (B. 5. def. 11). 

Q^ E. D. 



BOOK FI. PROP. XX. THEOR. 



243 




IMILAR poly- 
gons may be di- 
vided into the 
fame number of 
fimilar triangles, eachfimilar 
pair of ivhic/i are propor- 
tional to the polygons ; and 
the polygons are to each other 
in the duplicate ratio of their 
homologous fides . 




Draw 



and 



and 



and " 5 refolving 

the polygons into triangles. 
Then becaufe the polygons 



are limilar, 
and — 




■■■•■■««««■ 



and 



♦=♦ 



are fimilar, and ^^ ^ ^J 
(B. 6. pr.6); 



but ^F^ = w becaufe they are angles of fimilar poly 
gons ; therefore the remainders ^^ and ^k 

hence nmmmmmmmmm * ■■>..■«•• \\ -_..._-__ * 

on account of the fimilar triangles, 



are equal ; 
* ? 



244 BOOK VI. PROP. XX. THEOR. 
and --. : :: I 



on account of the fimilar polygons, 

■ ■■■■*■■■■» • _^.^_^_— — ___ •• ■■■■■■■HMM* • _^ 

ex asquali (B. 5. pr. 22), and as thefe proportional fides 
contain equal angles, the triangles 



s ^^^ and ^^^ 



are fimilar (B. 6. pr. 6). 
In like manner it may be fhown that the 

triangles ^^F and ^^K are fimilar. 



^^ and ^^m 



But -^^» is to ^^^m in the duplicate ratio of 
..-■■..... to .—.—>— (B. 6. pr. 19), and 



^^^ is to ^^ 



in like manner, in the duplicate 
ratio of -.■•■•■■-.• to —-——.; 



>> 



(B. 5-P'-. II); 
Again ^^^^ is to ^^^ in the duplicate ratio of 



^^^ to ^^ 




to — ^— — , and ^^^F is to ^^r in 



T 



BOOK VL PROP. XX. THEOR. 245 

the duplicate ratio of ^i^— to . 







and as one of the antecedents is to one of the confequents, 
fo is the fum of all the antecedents to the fum of all the 
confequents ; that is to fay, the fimilar triangles have to one 
another the fame ratio as the polygons (B. 5. pr. 12). 




But ^^M is to ^^^F in the duplicate ratio of 

to 




Q ED 



246 



BOOK VL PROP. XXI. THEOR. 




ECTILINEAR Jigures 



( 




<?«</ 





which are fimi/ar to the fame Jigure ( 
are fimilar alfo to each other. 




Since HHiBll^ and are fimi- 

lar, they are equiangular, and have the 
fides about the equal angles proportional 
(B. 6. def. i); and fince the figures 




and '^%. are alfo fimilar, they 
are equiangular, and have the fides about the equal angles 





proportional ; therefore IHIBl^ and l^Hhk. are alfo 
equiangular, and have the fides about the equal angles pro- 
portional (B. 5. pr. 1 1), and are therefore fimilar. 



Q,E. D. 



BOOK VI. PROP. XXII. THEOR. 247 




PART I. 

Y four Jlraight lines be pro- 
portional (^^^ I ^^— 

:: — : ), the 

Jiinilar reSiilinear figures 
fimilarly described on them are aljo pro- 
portional. 

PART II. 

And if four fimilar reSlilinear 
figures, fimilarly defcribed on four 
jlraight lines, be proportional, the 
firaight lines are alfo proportional. 






Take 
and — 



to 



fince 



PART I. 

a third proportional to 



, and —■••••■•• a third proportional 
— > and — — — > (B.6.pr. ii); 

:: ; (hyp.), 

■ — ■ :: -— — : -■••••••••• (conft.) 

.*. ex asquali. 



but 




and 




(B. 6. pr. 20), 





248 BOOK VI. PROP. XXII. THEOR. 




and ,*. 






(B. 5. pr. 11). 



PART II. 

Let the fame conftrudlion remain 







(B. 5. pr. II). 



(hyp-). 

(conft.) 



(^E. D. 



BOOK VI. PROP. XXIII. THEOR. 249 



QUIANGULAR parallel- 
ograms ( and 

^m^ ) are to one another 
in a ratio compounded of the ratios of 
their fdes. 




Let two of the fides 



and 



-«... . about the equal angles be placed 
fo that they may form one ftraight 
line. 

Since ▼ 4. M -- f\\ , 

and 1^^ = ^W (hyp.). 




and .*. 



+ 

and 



form one flraight line 



(B. I. pr. 14) ; 
complete ^ , 



Since 




# 



• ■■«■• 



and 



# 



(B. 6. pr. i), 



(B.6. pr. i). 




has to 

- to .„. 



a ratio compounded of the ratios of 
, and of ^^— — to — n^— »» . 



K K 



Q^E. D. 



250 BOOK FT. PROP. XXIV. THEOR. 




-B 



N any parallelogram ( ^7 / ) 
the parallelograms ( r^i 

and ^ I ) 'which are about 
the diagonal are Jimilar to the whole, and 
to each other. 




B-J ^^ 



As ^ I and ^ I have a 
common angle they are equiangular ; 



but becaufe 



and 



are fimilar (B. 6. pr. 4), 



and the remaining oppofite fides are equal to thofe, 

, It 1 and fn I have the fides about the equal 
angles proportional, and are therefore fimilar. 

In the fame manner it can be demonftrated that the 

parallelograms ^7 / and ^ / are fimilar. 
Since, therefore, each of the parallelograms 



B ..^E 



^. 



is fimilar to ^1 I , they are fimilar 
to each other. 

Q. E. D. 



BOOK VI. PROP. XXF. PROB. 



251 




O defcribe a reSlilinear Jigure, 
ivhic/i /Jiall be Jimilar to a given 



reBilinear Jigure ( 
equal to another (^^ ). 



), and 



Upon defcribe 

i_ defcribe | | = ^^, 



and upon «_ 

and having ^M ^ 




(B. I. pr. 45), and then 



smm 



Between 



and 



■H»» will lie in the fame flraight line 
(B. I. prs. 29, 14), 

and nu»»H.. find a mean proportional 
(B. 6. pr. 13), and upon _^««_«i 



defcribe Jtt^ 9 iimilar to 
and fimilarly fituated 



Then 



For fince 



and 



are fimilar, and 

(confl.), 



■ ■■■■■■■■■ 



(B. 6. pr. 20) ; 



252 BOOK FI. PROP. XXV. PROP. 



but 1 




• • 








but .^d^k = ■ 



(B.6.piM); 



(B.5.pr.ii); 



and .♦. 



(conft.), 
(B. 5. pr. 14); 



I 



and 



(conft.) ; confequently. 



which is limilar to 



is alfo = 



Q. E. D. 



BOOK VI. PROP. XXVI. THEOR. 



253 




F fitnilai' and Jimilarly 
pojited parallelograms 



have a common angle, they are about 
the fame diagonal. 



For, if poffible, let 



be the diagonal of 
draw ■ 





(B. I. pr. 31). 



Since 



P.. ^ 



are about the fame 



and have 



diagonal ^^^^^^^^ , ana nave jmm common, 
they are fimilar (B. 6. pr. 24) ; 



but - 



(hyp.). 



and .*. 



(B. 5. pr. 9.), 



which is abfurd. 



3 



is not the diagonal of 
in the fame manner it can be demonftrated that no other 
line is except : . 

Q. E. D. 



254 



BOOK VI. PROP. XXVII. THEOR. 





F al/ the reBangles 

contained by the 

fegments of a given 

Jlraight line, the 

greateji is the fquare which is 

defer ibed on ha f the line. 



be the 



unequal fegments, 
equal fegments ; 



For it has been demonftrated already (B. 2. pr. 5), that 
the fquare of half the line is equal to the redlangle con- 
tained by any unequal fegments together with the fquare 
of the part intermediate between the middle point and the 
point of unequal fection. The fquare defcribed on half the 
line exceeds therefore the redtangle contained by any un- 
equal fegments of the line. 

Q.E. D. 



BOOK VI. PROP. XXFIII. PROP. 



^SS 




O divide a given 
Jlraight line 



fo that the rec- 
tcuigle cojitained by its segments 
may be equal to a given area, 
not exceeding the fquare of 
half the line. 

Let the given area be := 
Bifedl — 




or 



make 
and if 



But if 
muft 



■ ■■■ 




» **** ymmmm*** 


■■■1 


•; 

. 2 


the 


problem 


is folved. 




9 




■II ^ 


-4;- ••■»■■■■••■ 


9 


then 


■■■ 





(hyp.). 



Draw 

make - 
with ^-i 



■ ■■■■IMHM I 



or 



as radius defcribe a circle cutting the 



given line ; draw 



Then •••— ^ wiMMaBB.-a- ^ 

(B. 2. pr. 5.) = 



.2 



But 



+ 



(B. I. pr. 47); 



256 BOOK VI. PROP. XXVIII. PROB. 



.\ X — ■ + ^— 

= ' + \ 

from both, take — i"— ^— ^^ 
and ""■■ X ■— — ^^—•■••« S3 «MMB 



But " ■■ ' ■ ' =: — — — •• (conft.), 

and /. ■-■"" is fo divided 

that •"•"• X ——————— ^: —•••.-2^ 

Q^E. D. 



BOOK VI. PROP. XXIX. PROB. 



^S7 




O produce agivenjlraight 
line ( ), fo 

that the reBangle con- 
tained by the fegments 
between the extremities of the given 
line and the point to which it is pro- 
duced, may be equal to a given area, 
i. e. equal to the fquare on 




Make 



draw -—"--■ 
draw 
with the radius 
meeting 

Then —■■-—' 



•-, and 



But 



and 



5 and 

', defcribe a circle 

■ produced. 



' (B. 2. pr. 6.) = — 



-^ + 



(B.i.pr.47.) 




:= the 



given area. 



Q^E. D. 



L L 



258 



BOOK VI. PROP. XXX. PROB. 





1 








O cut a given finite Jlraight line ( ■ — ■• ) 
in extreme and mean ratio. 



On 



defcribe the fquare 



I 



(B. I. pr. 46) ; and produce 

X " 



, fo that 
^ s 



(B. 6. pr. 29); 



take 
and draw ^ 
meeting ^ 



Then 



U 



■•aatB •■ 



X- 



;B. I. pr. 31; 



■ ■■■taaBB 



and is .*. ^ 



n 



; and if from both thefe equals 



be taken the common part 



\ I , which is the fquare of ■ 

will be = ■ , which is = '■■- X 



■ ■««■■«■*« * 



that is 






and 



.■ is divided in extreme and mean ratio. 

(B. 6. def. 3). 

CLE. D. 



BOOK FI. PROP. XXXI. THEOR. 2 



59 




F any fimilar reSlilinear 

figures be fimilar ly defer ibed 

on the fides of a right an- 

gled triangle ( ^''•»^ ), the figure 

defer ibed on the fide (■....i ) fuh- 

tending the right angle is equal to the 
futn of the figures on the other fides. 




From the right angle draw 

to •> m. 

then ■■■■■•■■■MHB : ^_— i— 



perpendicular 



but 



(B. 6. pr. 8). 



(B. 6. pr, 20). 



(B. 6. pr. 20). 



• •ammmmmm%mmm 



Hence 



but 



+ 




+ 



and /. 



Q. E. D. 



26o BOOK VI. PROP. XXXII. THEOR 




F two triangles ( ^ ^ ^«^ 

/^\ ), have two fides pro- 
portional ( ..1.^.^^ : I I 
\\ .. .......... I •.•••..•••.), and be fo placed 
\ i7^ an angle that the homologous Jides are pa- 




rallel, the remaining Jides ( 
one right line. 



and 



) form 



Since 



= (B. I. pr. 29); 

and alfo fince -^— ^ || ••■••••■>■ 

= ^^ (B. I- pr. 29); 
= ^^^ ; and fince 



■ ■■■«■■■««■ • 



— (hyp.). 



the triangles are equiangular (B. 6. pr. 6) ; 



M = /S 



but 



A+ +A = 



+ 



A 



+ JI = 



■*■«■»■»« 



I I 1 (B. I. pr. 32), and /. -*••—■» and 

lie in the fame ftraight line (B. i. pr. 14). 

Q,E. D 



BOOKVL PROP. XXXIII. THEOR. 261 




N egua/ circles ( 



O-O 



), angles. 



whether at the centre or circumference, are 
in the fame ratio to one another as the arcs 



on which they Jland ( 
fo alfo are fedlors. 



i-J::- 



o 



)■> 



Take in the circumference off 1 any number 

of arcs "■— ■ , ■— , &c. each ^ ^m» ^ and alfo in 

the circumference of f j take any number of 

arcs • , , Sec. each ^ •***«*•, draw the 

radii to the extremities of the equal arcs. 

Then fince the arcs — , —. , i..., &c. are all equal, 
the angles # , # , ^, &c. are alfoequal (B. 3. pr.27); 

.*. ^V is the fame multiple of which the arc 
is of ^1^ • and in the fame nianner ^Bi^ 




is the fame multiple o 
is of the arc 



which the arc 



.... •>•• 



V* 



262 BOOK VI. PROP. XXXIII. THEOR. 
Then it is evident (B. 3. pr. 27), 

if" ^11^ (or if m times w ) C> => ^ Mfg^ 



I 



(or n times ^ ) 
then ^^fc_i,i«^^ (or »; times '••••^) C!> ■ 
.....^••* (or n times ) ; 






.... , (B. 5. def. 5), or the 
angles at the centre are as the arcs on which they fland ; 
but the angles at the circumference being halves of the 
angles at the centre (B. 3. pr. 20) are in the fame ratio 
(B. 5. pr. 15), and therefore are as the arcs on which they 
ftand. 

It is evident, that fedlors in equal circles, and on equal 
arcs are equal (B. i. pr. 4; B. 3. prs. 24, 27, and def. 9). 
Hence, if the fedors be fubftituted for the angles in the 
above demonftration, the fecond part of the propofition will 
be eftablifhed, that is, in equal circles the fedlors have the 
fame ratio to one another as the arcs on which they ftand. 

Q^E. D. 




BOOK VI. PROP. A. THEOR. 

Y the right line {'mmm^um,), 
bifeSling an external 

angle ^H of the tri- 

yf 



263 



angle 



z. 



meet the oppojite ^ 




Jide (-^^— •) produced, that whole produced fide ( "■■•), 

and its external fegment (——--—) will be proportional to the 
fides (-^— — ■..— and ), which contain the angle 

adjacent to the external bifeSled angle. 

For if I be drawn || -.---»•»■• ^ 

then ^^ = \ / , (B. i. pr. 29) ; 



and 



= ^,(hyp-). 

= ^P, (B. I. pr. 29); 
r........ zz. ■ I III. , (B. I. pr. 6), 



and 



(B. 5. pr. 7) ; 
But alfo. 



■ •■■■■■■■■■■ , 



(B. 6. pr. 2); 
and therefore 

(B. 5. pr. I,). 



Q. E. D. 



264 



BOOK VI. PROP. B. THEOR. 




X 




F an angle of a triangle be bi- 
Je5ied by a Jlraight line, which 
likewife cuts the bafe ; the rec- 
tangle contained by the Jides of 
the triangle is equal to the rectangle con- 
tained by the Jegments of the bafe, together 
with the fquare of the Jlraight line which 
bifedls the angle. 



Let 



be drawn, making 
^ = ^; then fhall 

X + 



^r \ defcribe I 



About y \ defcribe J (B. 4. pr. 5), 

produce 1 ■■ to meet the circle, and draw ■■■»>— 

Since ^ = ^^ (hyp-)' 
and ^^ = ^ (B. 3. pr. 21), 



• * •C*i»l»l 



ind 



\ 



are equiangular (B. i. pr. 32) ; 






(B. 6. pr. 4) 



ROOK FI. PROP. B. THEOR. 265 

(B. 6. pr. 16.) 

.- X + ' 

(B. 2. pr. 3); 



■*«■«■ 



but X — — = X 

(B- 3- pr. 35)' 

X = X 



Q.E. D. 



MM 



266 



BOOK VI. PROP. C. THEOR. 




fhall 



Y from any angle of a triangle a 
Jlraight line be drawn perpendi- 
cular to the bafe ; the rectangle 
contained by the fdes of the tri- 
angle is equal to the reSlangle contained by 
the perpendicular and the diameter of the 
circle defcribed about the triangle. 




From 
draw ■«>ii»«afa«« 

.. X 




of ^y 



..-• 



— ; then 
Xthe 



diameter of the defcribed circle. 



Defcribe 



O 



(B. 4. pr. 5), draw its diameter 



and 



. and draw ^-im— • then becaufe 
^ ■ >• (conft. and B. 3. pr. 31) ; 

,Xl = /> (B. 3. pr. 21); 




.%*< 





is equiangular to / ^ 


• MHHaHMMHM 


l/^ 


and ,*. .-— .— 






(B. 6. pr. 16). 



(B. 6. pr. 4); 



X 



Q^E. D. 



BOOK VI. PROP. D. THEOR. 



267 




IHE reStangle contained by the 
' diagonals of a quadrilateral figure 
I infcribed in a circle, is equal to 
\ both the reBangles contained by 



its oppoftte Jides. 



/ /be any quadrilateral 
/ 

o 



fieure infcribed in 



and draw 




and 



then 



X 



X 



Make 



^k = W (B.i.pr. 23), 
^ = ^ ; and 




(B. 3. pr. 21); 



= 



«■■■■■■!■■« 



(B. 6. pr. 4); 



and ,*. 



X 



X 



(B. 6. pr. 16) ; again, 
becaufe ^^ ^ ^F (conft.), 



X- 



■■■■■•■•• 



268 BOOK FL PROP. D. THEOR. 

and\/ = \^ (B. 3. pr. 21); 



•■«■■■«•■■• * ■■■■■*>■■■ 



THE END. 



(B. 6. pr. 4); 
and ,'. '•"••■-••• ^ .^__^ ^ .•■■•■•■■■■• ^ a^^^MB 

(B. 6. pr. 16) ; 
but, from above, 

X = X ; 

— = X + X 

(B. 2. pr. I . 

Q^E. D. 



cHiswirK: PRiNirn by c. " iirxTiNoinM. 









fuciJ 



■O.,' 



^ \'yj